Well, I already got some 16's that got 205's on 'em. I want to run 215's or maybe even some 225's. Now, besides a wider wheel and tire, would there be anything else I need? Would running wider wheels/tires affect the suspension, or would I be safe as long as the offset is the same?

If you get wider wheels and tires you are going to run into clearance issues. You didn't mention what kind of car you have how wide your wheels are now and what offset. I run a Civic and had 16x7.5" wheels with 43mm offset and 215/40x16" tires and had the inner lip of the wheel rubbing my rear upper control arms.