I was in your same position 2 years ago. I went to a Precision Bicycle out on sacramento and picked up a brand new Schwinn Le Tour (last years model) for $500. Sound’s like a lot, but was pretty much one of the best investments I ever made. And besides a couple water bottles, tubes and a portable pump, it was all I ever had to by. Granted, I did get into it a little bit more and ended getting some nice clips and shoes later, but if you’re not doing 30 mile rides they really aren’t necessary.
If you’re going to be riding out of town (back-road as you say) then you might want to look into some hybrid tires too. Idk what the roads you’re planning to ride on look like, but standard no-tread road tires will pop tubes all day if there’s cracks and potholes everywhere.
As for size, I would just walk into a bike shop (berts, precision, etc) tell them you’re interested, learn a few things, and have them test fit on some.
I would recommend talking with Tom at Tom’s Pro Bike Shop, or the guys at Handlebars.
Different frames from different manufacturers, even in the same size vary in how they fit different body types. You will want to be sure you are on the right size bike. Just because you are 53cm on one bike, does not mean it’s going to be the same size on the next bike.
Either of these places will fit you on the bike to be sure it is the right size, and adjusted correctly for you.
You can get a decent entry level road bike at either of these places for around $700-800.
I agree on all accounts with Toms Pro Bike. My GF went in there to buy me a trainer for Christmas one year and they didn’t talk down to her like some bike store snobs do. I’ve sent Jeller over there and he grabbed a Fuji I want to say. I also agree that different bikes are sized differently but even the biggest 51cm is going to be on the small size for someone who’s 5’9".