See if you can find a nice deal on a 700c bike rather than a 27" one (check the tyres if you're not sure). Another thing to look out for is how the brakes are mounted:





The blue bike is typical of either older bikes (before early 80s) or cheap and nasty ones. If a bike's nice, but has nutted brakes, that makes it truly vintage IMO.

You should be able to find a bike more like the red one for $250 or less. It should have a 7spd cassette hub, which is good, because among other things that means Hyperglide and indexed shifting.

So if you get a 700c bike with 7spd and brakes with the recessed allen head fitting, that'll be the most bang for your buck, and the most amenable to future upgrades/replacements. One thing worth doing straight away is to ditch the old front brake for a dual-pivot.