Here's what I did - not perfect, by any means, but it's working for me: Start at the thrift store. They've seen it all before. Figure out your sizes, colors and styles by trying and buying used clothes. Once you can eyeball a top or pair of pants and know that they're the right size, you can move on.
Not kidding. A trans person in the girls' clothes area at a thrift store is like a twenty-times-a-day event. Nobody will care, and it'll give you breathing room to realize that no matter where you are, nobody actually cares too much that you're in the women's section.
It does take time, and I still get nervous. But honestly, it's something we all make far too much of a meal of.