I would recommend a sports bra to start with. The sizing tends to be more flexible, and they are often sold in smal-medium-large, rather than numbered sizes. Sizing is by band size, not cup size. I needed a large even though I had no cup size to speak of.
A sports bra will protect your tender nipples while allowing room to grow. If you get one with tank-top straps, it makes good camouflage if you are still presenting as male, since people will assume a tank top if they see the straps.
Yes, going into the store is weird at first. The first thing to remember is that your money is as good as anyone else's, and the store wants it. The second thing is that you are not breaking any laws. You have as much right to be there as anyone else. You could be shopping for your wife or girlfriend. Just take a deep breath and do it!
It gets easier. I went bra shopping quite a few times in my cross-dressing days, and eventually it was no big deal. But, if you have an understanding female friend, it will be easier on your ticker to have her along for support.