A 20 gallon (long) would have worked fine. What size is your new one?

I learned most of what I know about proper husbandry by looking at the caresheet and surfing various threads here. Whether or not you post directly, you'll find the people here to be immensely helpful. I think the most important thing, above all else, is providing the right temperature gradient: 80-84 on the cool side and 90-94 on the hot side (the temperatures vary by opinion but those are the general ranges you'll find). Generally speaking, these ideal temperatures are harder to reach in a larger tank (just a head's up).

I'd recommend purchasing a UTH (or two), an Acu-Rite thermometer (it has a probe so you can measure the hot and cool sides simultaneously. It also reads humidity. You can get one from Walmart for about $12) and two identical hides at the very least (one each for the hot and cool sides). You will also need a thermostat to control the UTH because they can reach really high temperatures. There's a variety available at RBI, and you can buy several other supplies from there.

If you're sticking with a tank, I'd treat your cage top with some foil to help with the humidity issue. For my 20 gallon long, I have 3/4 of the screen top covered with three sheets of foil (shiny side down) held down with duct tape. I still spritz every few days but the humidity stays right where it should be (around 50-60%). Prior to treating my tank, my BP had decent sheds but they came off in several pieces. Since I've modified it, he's shed once and it was absolutely perfect. (Thank-you to Argentra for this idea.)

Note that your heat lamp is what's likely sucking out most of the humidity as well. I heat the hot side of my BP's tank with a UTH on a Johnson A419 and the cool side with a 60 watt red bulb... so it's possible to use a heat lamp. I wouldn't say it's the best set-up (I personally want to spring for another UTH and thermostat when I can) but it can work. A lot of people like to put their heat lights on a rheostat for ideal control.

So for your shopping list, I'd add:

- 1 or 2 UTH (depending on whether you keep the heat lamp to heat one side)
- Thermostat
- Rheostat
- 2 Identical Hides (if you don't already have them)
- Acu-Rite
- Foil
- Duct Tape

That's just my two cents. I'm sure I missed something and anyone is welcome to correct/add to what I said. I hope it helps. Good luck!