Tips for shooting in public?

I’m an illustrator so I usually work alone, by myself, in my house. I love photography though, and right now I mostly just take and edit etsy product photos, or abandoned buildings in my hometown. I live in what I think is the most beautiful part of a major historic city, and I want to try my hand at street photography, or even just nature photography at the park, but I feel awkward just thinking about it. How do you get past the initial nervousness? Is it better to be obvious with a big camera, or discrete with a little one? Any and all advice is appreciated.

You can start photographing w/ you cell phone until you get over your shyness. Most folks these days think nothing of see people photographing w/ one and they can take surprisingly good street photos.
Then you can graduate to a quality discrete fixed lens mirrorless camera like the Ricoh GRIII, Fuji X100V, Nikon ZFC w/ kit lens and models by Panasonic, Sony, and Leica. These are all high quality discrete or retro cameras that don't seem to offend subjects.
Street photography takes practice but the results are rewarding.

+1 iPhone is a great 

I don't know what camera you have but a mirrorless or compact camera might be a good starting point. If you're an illustrator you may have enough art background to be familiar with rangefinders and their history and use in enabling photographers to take candids without being as noticeable.

It probably takes trying it and seeing if it's for you. If you like photographing architecture that might be a good starting point into street photography. This came up in another thread, but I've done home visits for the type job I had in a major city and was in many neighborhoods that are often the type where people do street. I think you need to keep in mind that's where they live, their neighborhood, and be respectful (not that you wouldn't, I don't know you, and you sound like you would be). It just came up in another discussion and maybe I misunderstood comments and was mistaken but it seemed like some didn't think it was all that necessary as a photographer to treat people with respect no matter where they live. But I think it is...

Anyway, depending on the neighborhood, be aware of personal safety. I didn't encounter it often but there are obviously people on the streets dealing with mental illness, poverty, drug use, etc. I found that if people see you around and get to know  you they may become comfortable with you.. Initially they may be apprehensive about why you're there.

I've found that being a photographer often involves interacting with people. I've done sports and events, but otherwise I'm photographing buildings or nature etc. and I'm just as glad if people stay out of my viewfinder!! But still I try to nod, smile, say hello. 

Just a thought - maybe take a small sketch pad with you, one you've used (not your best work that you wouldn't want to take a chance losing/dropping etc.); do a quick sketch or two along with taking photos so people who see you get an idea what you're doing.