First of all, welcome aboard. I think you'll find this board to be an invaluable resource. And welcome to the fraternity. Refereeing is a very challenging and very rewarding avocation.
As to your question, you may want to ask around in your local area. In my area of the country, people wear the crew neck T under the stripes. However, it may be a be a bit of a faux pas in your neck of the woods.
It sounds like your first game went about like most people's first games
Your partner is correct. Pick a couple of things to work on. May I suggest you pick the ones that will make the biggest difference first? The self-improvement part never ends, so just get used to it. And, if you have already gotten a mentor, you're off to a good start.
You may feel a little awkward and sometimes clueless out there at first. That's normal. You may even have games when you'll be glad to just get through. That's normal too. But with each game you'll become more comfortable.
At some point, you'll have to give your first T. The reason I bring it up is that it can be a rather traumatizing experience for some new officials. But it's just business and don't let it ruin your game when it happens.
If both coaches had positive things to say, then you probably did okay. Having said that, don't let a coach, player, or fan be your measuring stick. But they can be something of a barometer to let you know when you're really stinking it up (and we ALL have one of those games occassionally).
Keep working hard. Solicit feedback from your partners and anybody else who is qualified to give it. Don't let negative feedback get you down. And go to a good camp this summer.