What should I be doing in the boxing gym

The ONLY thing you should be doing is finding a gym where you will have a coach. Everything else is fitness boxing.

You should be boxing.

Find a gym that has competing amateurs. Go at least 3 times a week. Spar as much as you can with as many different partners as you can. Try to run a few days a week to start, if your really serious run every day. Your coach should start taking you seriously once he sees your there lots and putting in the work. Adult boxing class to me sounds like "boxing fitness" and not training to fight in the ring. If this is true its probably about as useful as going for a run but nowhere near as effective as the training most amateurs do daily. Tell your coach you want to fight after you've shown your coach you want to fight. The minimum work the amateurs I train with put in is 3 hours a day, 3 days a week at the gym and 10+ miles a week running. The serious fighters do alot more than that. If you've been training 4 months already it shouldnt take you long to get ready for your 1st fight. At our gym most comitted fighters have there 1st fight within 3 months to a year of starting.

You should be carrying spit buckets and pulling pubes out of the soap in the showers. Haven't you watched the Rocky movies?

Ask the coach to train you silly.

Go to a dedicated Boxing gym that's full of fighters, thats where you will learn.

Probably the wrong section for this...expect to get trolled by a bunch of fat idiots who have never boxed competitively. Edit: see above post for proof