I'm a lot taller than you, so I can't help on bike size.

If you're looking for a car replacement, pretty much any kind of bike that isn't a super-light racing only bike will work. You will want fenders and a rain jacket, so that if it rains your wheels won't get you all muddy and you won't get soaked. You will want a headlight and taillight, so riding in the dark or fog won't be as dangerous. You'll also want reflective tape (for your fenders and helmet), and maybe some more reflectors. You may want a messenger bag, backpack, bike rack, a front basket, a rear basket, or wire or cloth panniers for carrying things.

Helmet and gloves are safety features. A cracked head or roadburned hands are not good. Glasses too, eyeballs are expensive.

If you plan to ride for more than about 20 minutes, you'll want a water bottle and a way to carry it. I keep mine in my backpack.

A lock (or two) so people can't walk off with your bike. A kickstand if your bike didn't come with one. A pump and spare inner tube so that when you get a flat, it's not the end of the world. A mirror can help in traffic.

Most bikes do not come with anywhere *near* what they need to be an effective car replacement. Plan for this in your budget. Depending on the bike you find, you may need to add nearly all these accessories right away.

I'm finding that a backpack I already had is doing fine for grocery duty. It's a bit hot on my back, but livable. I wear glasses, so I didn't need to buy special ones (and if I get tired of blowing dust, I'll start carrying my lab goggles with me). Cheap helmets are just as good as expensive ones at keeping your head intact. A lock is worth a bit of a splurge. Lights and reflectors are also worth a bit of a splurge. Don't go nuts, but a light with lots of LEDs tends to be easier for drivers to see and easier for you to see with.