In most cases, at big airports you will have the tower's attention right away if you execute a go around on your own; usually before you could blow through your initial climb out altitude. Usually, you will get an even IFR altitude, not a VFR one since you are still on an IFR flight plan and have to be vectored with other IFR traffic. I have however once seen a WN 737 get assigned 1500 at LGA initially to avoid a chopper over the field just as they went around. I recently did a go around from rwy 22 at LGA and was given 3000 and a right turn out over central park and then the hudson river to get vectored back around. So I would say procedurally do whatever your company and the AIM say as far as what altitude to set, but try to announce the go around as soon as you know you're going to do it and listen for ATC instructions.