For what it's worth, my first ventures out in public were at 5:00 a.m., in the winter, when it's dark. Every now and then a car would pass, but despite my fears, nobody jumped out to beat me up. I get my mail at a post office box, and I discovered that they leave the lobby area open all night, so then I added dropping off and picking up my mail. One day the postmaster (postmistress?) surprised me when I was wearing a long red satin skirt, and she just said it looked nice.
I eventually graduated to walking down to the supermarket when it opened, at 6:00 a.m. (still before dawn), and nobody even looked at me funny.
Okay, this is in the New York City suburbs, so many people have seen a thing or two already, but I think people are mostly just so busy with their own lives they don't have time to worry about someone who looks a little different. If they even notice.
The other advice is to practice convincing yourself there's nothing odd about how you're dressed, if they don't get it, they're just ignorant or uninformed. I mean, you could be wearing a clown outfit, but if you knew you were on your way to be the clown at a kids' birthday party, you would just shrug at the funny looks. Guy going to entertain at a party, stops in to pick up a 6-pack of beer for later, what's the big deal? Oh, and here's my business card.