It’s not as involved as you might think. I’ve actually had 2 go arounds in 2 weeks. First one, we were trying to make curfew but had everything going against us. Second one was a windshear warning at 500ft in SYD.
Well, as you alluded to, pushed the thrust levers forward and gained some altitude, got a positive rate and retracted the gear. Next we retracted the flaps, made a left turn to join the circuit (usually just one continuous turn), completed the after take off checklist, made a PA to explain why we went around. Do the descent and approach checklists, another turn onto base, start slowing up again and taking flaps as normal. Start the descent at the same time and when ready drop the gear and do the landing checklist once all the flaps are out. One last turn onto final and line her up. Conduct the best landing ever and have everyone clapping.