My strategy:
1) thrift stores
2) department store mens' sections
3) walmart or target
4) internet
Thrift stores are awesome because no matter how odd you are, you're nowhere near the oddest person there. The employees there see everything, and are usually very tolerant. People also shop for costumes at thrift stores a lot, so you can always pretend (either to tell people or just imagine in your head) that you're buying for a cross-dress party or something. Easier in October than November, but these things happen year round. If you can't try clothes on, it's usually $1-5 per article, so it's a minimal risk to take it home and try on there.
Department stores and discount stores are big enough that even if you get the hairy eyeball, nobody will remember an hour later. Unless you're in a very small town in a conservative area, you'll be only one of many XX-chromosomal people shopping in the men's department. You can even pull men's stuff off the shelf and go try it on in the women's dressing room. The first time around, I felt paranoid and grabbed some women's articles, too, to cover it up. But nobody even noticed what I took into the dressing room.
I did the first two steps to get a sense of sizing in different brands. Men's clothes are also great because sizes are actual measurements, not some arbitrary number. Sizing isn't 100% consistant, but you can get a decent sense of your shirt size by measuring your neck and chest circumfrence, and arm length (usually measured from the nape of the neck to the wrist). Most shirts are sized neck x arm, but the manufacturer's website will usuaally give you chest circumfrence. Especially if you're not large-breasted, or you're binding, men's sizing is fairly reliable. Once you've found a brand you like and that fits you, go with it. Nothing wrong with a boring wardrobe when you're starting out.
It's also very common for women to buy clothing for the men in their lives. No one bats an eye a women buying men's clothes in most places. My dad used to want socks and underwear for Christmas every year. Its also not uncommon for women to wear men's clothing, especially jeans and shirts. FtMs are lucky in this. Trans women have trouble accessing wardrobe basics unless they're passing.
The other thing to remember is that you're much more aware of yourself than the people around you are. Like I said, even if you get a wierd look, it's unlikely that anyone will remember unless they know you fairly well. Feeling nervous will attract more looks than the clothes themselves which, like I said, there are lots of reasons for even a cis woman to be buying.
