Thanks for the pix, meerkat! You know, your bike looks a lot better than I thought it would.
What GuitarStv said. He posted while I was still typing my reply, but I'll post anyway.
If it were mine, I would replace those brake cables... it's really easy and cheap. Basically a case of removing the old ones, checking the housing for cracks and ragged ends, buying new cables and new housing if needed, then greasing the new cables (though I hear modern brake cables don't need grease), threading them through where the old ones used to go, and lightly attaching to the calipers (looks like you need a hex wrench.) Then adjust the length to make sure that the brakes work with a comfortable squeeze of the lever, give the hex wrench another couple of turns to make sure the brake is well attached, and finally snip off the excess cable. There are a lot of tutorials on YouTube of how to change brake cables... look and see if you think you could handle it.
Then I'd give those gears a scrub and lube... they really don't look too bad.
I'd also look at the ends of the chain and see if it is really broken (twisted or cracked), or whether the chain could be reattached to itself. If it could be salvaged, I'd clean it off and lube it a little, making sure each link moves smoothly.
Whether or not you can reuse the old chain or must get a new one, here's a good video on replacing a chain:
https://www.youtube.com/watch?v=rWchudX-TqsI wouldn't trust that your old chain was the right length (might be missing a link or two) so I'd use his method for finding the optimum length at 3:06 on that video.
Both of those fixes are easy and cheap.