constant movement of a game object along x axis

I am creating a 2D game in which the player needs to jump onto different platforms or fall onto another and stay alive as long as possible. So, the character needs to move along the x axis and be applied to gravity. Also I would like to add a slow but constant speed multiplier. This is a lot to ask for but I never made a game before and my scripting skills are at novice level. Can anyone at least point me in the right direction?

Since you’re using gravity (= physics), it’s always advisable to stick to the control you can have from the RigidBody itself.
It’s very bad practice to translate rigidbodies.

So in your case, if you want to constantly move an object on the x axis, simply give him a velocity (knowing there are no drag values on the rigidbody) and he will keep on moving:

this.rigidbody.velocity = new Vector3(mySpeed,0,0);

where mySpeed is your variable that you can increment it slowly based on your multiplier (control it via time, or level, or score…)

i know its late but this may work to someone (c#)

// public to change value on editor.	
       public float speed;
// this is for storing your rigidbody use RigidBody2D for 2D game
	private Rigidbody object; 
	void Start () {
 //setting the value, use <Rigidbody2D> for 2D Game
		object = GetComponent<Rigidbody> ();
        void Update () {
// moving the object on x, you can choose direction (x,y,z) change to vector2 if its 2D (x,y)
		object.velocity = new Vector3 (speed, 0, 0);