Should I just get my parents to mail the IRS forms from my old address to my new address (when the forms finally come), and then everything will be taken care of?
Or are there more steps I need to take care of?
IRS doesn't send the forms any more, so you don't need to worry about that at all.
You'll need to check carefully what's your tax liability in each of the states, and might end up filing returns (either resident or partial/non-resident) in both. The IRS itself doesn't care what income comes from what state (its a Federal agency), but the states do.