I was my object to move with a Vector3.right velocity, but moves like 0.0f 0.1f 0.2f 0.3f and on the same time, I want it still be able to check for collisions, as it ignores them if I moved the object manually with transform.position = x;
Consider using ForceMode.Acceleration to control your movement. Also multiply by Time.deltaTime for more precision.
rigidbody.AddForce(Vector3.right * Time.deltaTime * SomeSpeed, ForceMode.Acceleration);