No, that's not right at all. Your first assumption that only the elevator moves is the correct one. Some flyers do choose to incorporate flaperons that are coupled to the elevator function but this is only found on some serious aerobatic models.

If the ailerons are moving at all due to you inadvertently moving the stick while moving in the elevator direction then the ailerons should be moving one up and one down. Not both down together or both up together. Also if you carefully move the stick to move in the elevator only axis then at most you should just get a slight nervousness or very small wiggle of the ailerons where they hardly move at all. If it's more than that I'm going to guess that you have a fancy radio set that has some mixing functions. If so then the first thing to check is to plod through the programming manual and find out how to disable ALL of the mixing functions. By the sounds of things I'm betting that your Tx is set up with a mix or two in the "on" setting. For a trainer you definetly want to disable these.

The other thought is that if this only happens when you assemble the model that the elevator servo or pushrod is jamming into the aileron servo and pushing it. If this is the case you MUST correct this issue as the mechanical jamming of this sort will overload the servos. At best it would just result in using up your airborne battery sooner. At worse it'll result in the servo(s) overloading and failing due to the extra heat.