Can i switch my sim card from my iphone to a moto g?

I currently have an iphone 4 on an att gophone plan. It was my moms and once her contract was up she put it on my prepaid plan for me. Its gotten really slow and i want to switch to an unlocked moto g. Can i just switch the sim cards and be on my way or do i have to go to att and have them do it?

My current plan is the talk and text gophone plan without data and the iphone runs fine without data. At&t also said we don't have to purchase data on a smartphone if it is used on on the gophone plans, if it were on contract you do… So can i buy a no contract unlocked moto g and put the iphone sim into it? Will the new phone work or will at&t have to activate it?

Added (1). I do know that both the iphone 4 and moto g use a micro sim

Should be able to yes.

Surely, you can.