Sounds like you’re on the right track. You’ve got a good budget and the right idea. I’ve bought most of my bikes used, you can get great deals that way.
Find a good local bike shop and try bikes until you find one you like. I love my little Trek, but any of the brands they’ll have will be good. A lot of it just depends on what geometry is comfortable for you - only way to find that out is to do some test rides. They should be willing to let you take bikes out for a good spin.
Don’t let them talk you into suspension. Suspension is heavy and really unnecessary for commuter riding. Not to mention the maintenance, and hassle when it breaks.
You definitely want a helmet. Gloves are nice. I wouldn’t bother with bike shorts for a work commute, unless it’s really long. How far will you be riding?
A multi-tool for basic repairs is a good idea, plus a spare tube, and a little pack to carry them on the bike. Don’t forget to get a pump for the tires and some lube for the chain. Ask about dry lubes, they’re not as messy if you accidentally brush the chain. Oh yeah, a chain guard is nice, too!
You’ll need water, either cage + bottle, or a hydration pack. We just picked up a double-wall bottle with cage at Walmart for the SO, in hopes of keeping water cool for his commute in the 110 temps here. I dunno if he’s tried it yet, so I can’t report on the effectiveness. Get something with a mouth big enough for regular ice cubes.
You’ll probably end up wanting fenders at some point, so make sure those can be added later. Also a rack and panniers/baskets, you may want those someday too.
The SO keeps shoes, socks, and a set of clothes at work for unexpected rain events. He also has a rain suit, for expected rain events.
For heat, he just rides slow and takes it easy on the way in, to keep from getting too sweated up. Baby wipes are good for cleanup. But he gets to wear shorts and Hawaiian shirts to work, so that may not work so well for you. 
It also depends on your local temps, many locations you see people riding to work in suits. I don’t think that would work well here at the moment; our lows are in the 80s.
If you don’t want to carry clothes on the bike every day, maybe take in/pick up a week’s worth of clothes on Monday, and ride the rest of the week?