Actually I have some experience with this (yay for first post!) I have been fixing up a 65 monza convertible over the past year.
Parts are reasonable, Clarks (http://www.corvair.com/user-cgi/pages.cgi) will sell you about anything you need, as well as a few other venders. As pointed out above some things will be NOS or refurbs, but a lot of new or upgrade parts are available.
There are two great forums I've been relying on for information while fixing the car, corvaircenter and corvairform. Both places have people willing to help explain about anything and plenty of resources like links to videos on youtube or PDFs of manuals, etc.
I'd recommend a 65-69 unless you just happen to like the EM body style more. Also buy the one you find with the least amount of rust and the drive train in the best shape, those two will eat the most money as you can guess.
They're a lot of fun to drive and tool around in, not the fastest, but everyone loves them for how different they are. Have fun!
edit thought I'd add some things...
See if its leaking oil (likely) and from where. The o-rings for the push rod tubes are likely the cause or the oil pan gasket, both are easy and cheap fixes.
Also check the fuel pump (if it has mechanical) to see if it's new, if you have any fuel delivery problems to the carbs it's probably worn out and may be leaking fuel into the oil. This is also an easy and cheap fix unless you want to put in an electric pump.
Points on the car may also be bad if it's running poorly, you can replace those or put in something like an ignitor to replace the whole system and not mess with points.
Those are probably the main issues in a badly running corvair. If you can I'd suggest finding a 4 speed.
engines come in a few flavors, 110s are the most common i think and are good, 140s (corsas) which have 4 carbs and are a bit more finicky, and 180s (turbos) which have their own uniqueness.
I'd love either a 140 or 180 turbo for the coolness factor, but the 110 is a solid runner with not a lot that goes wrong on it once it's running.