Using older mountain bike in a Trainer
The point of the quick release skewer they give you with a trainer is generally to prevent damage to your fancy skewer already on your bike, and make sure the trainer can grip onto the axle properly. If you have a quick release system, you should use the provided skewer. 

If you have a thru axle like a Maxle, you may need to buy a special adapter. 

If you have a classic nutted axle (which you most likely have), you can generally use it directly in the trainer, but you may want to double check with the trainer manufacturer.