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Making Player Movement Less Floaty

So I am having issues with my runner game.
At the moment the movement feels way to icy because I have momentum, I have to stop going one way to go the other and it gives me this really icy effect I don't want.
I have tried upping friction but that results in my cube tumbling down the track
void FixedUpdate()
{
rb.AddForce(0, 0, forwardForce * Time.deltaTime);
if (Input.GetKey("d"))
{
rb.AddForce(sidewaysForce * Time.deltaTime, 0, 0, ForceMode.VelocityChange);
}
if (Input.GetKey("a"))
{
rb.AddForce(-sidewaysForce * Time.deltaTime, 0, 0, ForceMode.VelocityChange);
}
I want the movement to be nice and smooth and snappy.

So, first of all: I would recommend not using AddForce for movement of the player, you can just rb.velocity = sth; (that would remove the icy).
But! If you really want it that way, I guess you could try:
rb.AddForce(-rb.velocity*(from 0 to 1 the less the less icy) + (the speed you want to change to));
I think this would work. Try it, but also, if it's a capsule collider, you could loon at the Character Controller that makes it all look better and simpler, or at the Rigidbody Controller (not a component, just a name to define what you are trying but with the rb.velocity method).
Hope it works!

maybe this:
Vector3 CalculatedForce = -rb.velocity;
CalculatedForce.z = forwardForce * Time.deltaTime;
if (Input.GetKey("d"))
CalculatedForce.x += sidewaysForce * Time.deltaTime;
else if (Input.GetKey("a"))
CalculatedForce.x += -sidewaysForce * Time.deltaTime;
rb.AddForce(CalculatedForce, ForceMode.VelocityChange);

Unity: player moves up without any such code

I'm creating a movement function for my character(which I've done many times but this time it's not working correctly) and here's my code:
using System.Collections;
using System.Collections.Generic;
using UnityEngine;
[RequireComponent(typeof(CharacterController))]
[RequireComponent(typeof(Animator))]
public class Movement : MonoBehaviour {
public CharacterController controller;
public float moveSpeed;
public float jumpForce;
public float gravityScale;
public Animator animator;
private bool shiftPressed = false;
private void Update()
{
if (Input.GetKey(KeyCode.LeftShift) || Input.GetKey(KeyCode.RightShift))
shiftPressed = true;
else
shiftPressed = false;
if(shiftPressed)
{
moveSpeed = 20f;
} else
{
moveSpeed = 10f;
}
Vector3 moveDirection = new Vector3(Input.GetAxisRaw("Horizontal") * moveSpeed, 0.0f, Input.GetAxis("Vertical") * moveSpeed);
if(Input.GetKey(KeyCode.Space) && controller.isGrounded)
{
moveDirection.y = jumpForce;
}
if (moveDirection != Vector3.zero)
animator.SetFloat("Speed", moveSpeed);
else if (moveDirection == Vector3.zero)
animator.SetFloat("Speed", 0f);
else if (moveDirection != Vector3.zero)
animator.SetFloat("Speed", moveSpeed);
if(moveDirection != Vector3.zero)
transform.rotation = Quaternion.LookRotation(moveDirection);
moveDirection = Camera.main.transform.TransformDirection(moveDirection);
moveDirection.y = moveDirection.y + (Physics.gravity.y * Time.deltaTime * gravityScale);
controller.Move(moveDirection * Time.deltaTime);
}
}
As you can see that there is no function to go up in the y axis except the jump function. Mysteriously, when I press the 'S' key or 'downArrow' the player moves -z as he should but ironically he moves in +y axis as well. To ensure there is no y axis function I tried making jump function a comment but did the same way still. I thought it might be some character specific problem so I tried adding the code to a cube(thinking it to be my animation mistake) but it didn't helped at any point. I ensured the character controller is set nicely(collider and stuff); I've attached screenshots.
Please help!
Thanks in advance.

The problem has already been stated by several here, but it seems you still do not understand the issue, concluding from your reactions.
This line will always make your character move up.
moveDirection.y = moveDirection.y + (Physics.gravity.y * Time.deltaTime * gravityScale);
You are trying to use gravity (which is a force) in conjunction with manipulating transform. In Unity, you either do one of the two, not both. This will lead to undesired and hard to fix results.
If you want to use forces in your code, then i suggest the follow:
Add a RigidBody to your character. Check the "use gravity" checkbox.
Get the RigidBody in your controller script by calling and add force to it to move it.
var rb = getComponent<RigidBody>();
rb.AddForce(10f);
Do note that if you add force, you can add it continuosly in the Update method, or just once by passing a second paramater "forcemode".
rb.AddForce(10, ForceMode.Impulse); // add instant force, using mass
// Or
rb.AddForce(10, ForceMode.VelocityChange); // add instant force, ignoring mass
// Or
rb.AddForce(10, ForceMode.Force); // add continuous force, using mass
// Or
rb.AddForce(10, ForceMode.Acceleration); // add continous force, ignoring mass

Let me guess.. your character keeps moving up by about the current downwards slope of the camera?
As your code stands, as an example, if your Camera's is Vector3(10,0,0) and if you're trying to move Vector3(5, 0, 5), you've asked the movement vector to be transformed to world space, based off of the camera transform. So, your new movement vector will be something like Vector3(5,1,5). This should do the trick:
var worldDirection = Camera.main.transform.TransformDirection(moveDirection);
moveDirection = new Vector3(
worldDirection.x,
moveDirection.y + (Physics.gravity.y * Time.deltaTime * gravityScale),
worldDirection.z );

moveDirection.y = moveDirection.y + (Physics.gravity.y * Time.deltaTime * gravityScale);
This line always sets "y", and is modified/set as the last function prior to the ultimate statement. It is here, you should validate and verify all factors that has influence the final calculation of "y".
You must validate that each of the variables are calculated to your expected values:
moveDirection.y
Physics.gravity.y
Time.deltaTime
gravityScale
Each will influence the final value of moveDirection.y. Once you have verified each individual value, the final y value will make sense to you. If any of these values are not what you expect, you have a bug in the calculation of that variable.
Happy hunting!

Player object changes position while rotating

First of all: Sorry for my bad English, I'm Dutch.
I'm trying to create a player object that can walk forward and backwards (with the up and down keys), and rotate the player with the right and left keys. But when I press the left or right key, the position of the player changes, it looks like it rotates around a certain point. But it should rotate and stay on the same place, like I do when I turn around.
I have some other small programs, with the same 'move script' and the same inputmanager settings. There it works fine, so I have no idea why it doesn't work this time.
This is my move script, but this script works fine with other programs:
using System.Collections;
using System.Collections.Generic;
using UnityEngine;
public class MoveScript : MonoBehaviour
{
public float speed = 3f;
public float rotate = 50f;
private Rigidbody rb;
// Update is called once per frame
private void Start()
{
rb = GetComponent<Rigidbody>();
}
void Update()
{
if (Input.GetButton("Up"))
transform.Translate(Vector3.forward * speed * Time.deltaTime);
if (Input.GetButton("Down"))
transform.Translate(-Vector3.forward * speed * Time.deltaTime);
if (Input.GetButton("Right"))
transform.Rotate(Vector3.up, rotate * Time.deltaTime);
if (Input.GetButton("Left"))
transform.Rotate(-Vector3.up, rotate * Time.deltaTime);
if (gameObject.GetComponent<Rigidbody>().velocity.y < 0.01)
{
if (Input.GetButtonDown("Jump"))
rb.AddForce(0, 225, 0);
}
}
}
The InputManager settings (also the same as other programs where it works fine):
If someone wants a screenshot of something else of my program, you can always ask it of course. I have no idea what the problem could be, if it isn't in the script or the inputmanager.

Following the comments in the OP, the solution should be to use:
if (Input.GetButton("Right"))
transform.localEulerAngles = transform.localEulerAngles + new Vector3(xRotation, 0, 0) * Time.deltaTime;
if (Input.GetButton("Left"))
transform.localEulerAngles = transform.localEulerAngles - new Vector3(xRotation, 0, 0) * Time.deltaTime;
Where xRotation is a float with the amount you want to rotate per second when and while one of the rotation keys is down.
(PS: Don't have unity open right now, so + and - might be inverted, but should be something like that.)

Edit Script To Make Character Move In Direction He Is Facing

I am using the Third Person Controller asset by Opsive. It is a fairly complex third person controller that controls animations, damage, movement, inputs, etc etc.
http://opsive.com/assets/ThirdPersonController/documentation.php
I would like to update the ControllerHandler.cs script to make the character move in the direction that the character is facing, regardless of camera orientation. (In the style of old Resident Evil games)
In the original script, the character would move forward in relation to which direction the camera was facing.
I received some advice to change a line in this script, but with the changes, the character moves forward in one fixed direction. (E.G.: When I turn to the right with the D key, and then press W to move forward, the character turns back in the direction he was originally facing and moves in that direction.)
Here is the original part of the script:
#if ENABLE_MULTIPLAYER
if ( isLocalPlayer) {
#endif
if (m_Controller.Movement == RigidbodyCharacterController.MovementType.Combat || m_Controller.Movement == RigidbodyCharacterController.MovementType.Adventure) {
m_LookRotation = m_CameraTransform.rotation;
Here is what someone told me to change it to:
#if ENABLE_MULTIPLAYER
if ( isLocalPlayer) {
#endif
if (m_Controller.Movement == RigidbodyCharacterController.MovementType.Combat || m_Controller.Movement == RigidbodyCharacterController.MovementType.Adventure) {
m_LookRotation = Quaternion.Euler(PlayerInput.GetAxisRaw(Constants.YawInputName), 0, 0);
Unfortunately, this does not have the result that I intended.
Any assistance would be greatly appreciated. Thanks!
Here is a link to the controller script:
https://docs.google.com/document/d/1B4sstqtCqRMCLuHuxEuA9I7tO_3W4aHqEZwr73uFDjY/edit?usp=sharing

I think you want
transform.position += transform.rotation * Vector3.forward;
A complete block of code would look like this...
void Update() {
if (Input.GetKey(KeyCode.W)) {
transform.position += transform.rotation * Vector3.forward * MOVESPEED;
}
}

I have got it working. Take a look at the picture first it will help you set up the player correctly. My player is just two cubes. I added the second cube to give a face to where the player is pointing. I have added those 2 cube to a parent object and I move the parent object with my PlayerMovement.cs.
public float rotSpeed;
public float playerSpeed;
void Update()
{
if (Input.GetKey(KeyCode.RightArrow))
{
transform.Rotate( Vector3.up, Time.deltaTime * rotSpeed);
} else if (Input.GetKey(KeyCode.LeftArrow))
{
transform.Rotate(-Vector3.up, Time.deltaTime * rotSpeed);
}
if (Input.GetKey(KeyCode.UpArrow))
{
transform.Translate(Vector3.forward * Time.deltaTime * playerSpeed);
} else if (Input.GetKey(KeyCode.DownArrow))
{
transform.Translate(-Vector3.forward * Time.deltaTime * playerSpeed);
}
}

moving a rigidbody left and right

I can't seem to get my rigidbody to move left and right. The code looks fine and very similar to what everyone else has posted!
The debug statement is getting called but my character is not moving left and right.
Thanks for the help.
public float speed = 4.0f;
void Update()
{
float moveDirection = Input.GetAxis("Horizontal");
if (Input.GetKeyDown("d"))
{
Debug.Log("pressed d");
rb.AddForce(new Vector2(Time.deltaTime * speed * moveDirection, 0), ForceMode2D.Force);
}

I just tested it using an 3D environment, but that shouldn't matter. So after all I'm pretty sure you've got way to less force applied to AddForce.
So try increasing speed to about 40000, then you should be able to notice the AddForce being applied.
If you want to keep the speed value low, you could of course just add a multiplier here:
rb.AddForce(new Vector2(Time.deltaTime * speed * moveDirection * 10000f, 0), ForceMode2D.Force);

A nice one line option would be to use transform translate.
void Update ()
{
transform.Translate(Vector3.right * speed * Input.GetAxis("Horizontal") * Time.deltaTime);
}

AddForce will not work on a Rigidbody which is kinematic. Verify and set isKinematic to false in your Rigidbody component.
If this is already false, trying increasing the force value as suggested by d4Rk.