I did the Racer5 school, and then did their endurance races on their rented bikes for a couple of years. They're not doing endurance races anymore, but they have a couple of trackdays per year and will rent bikes for those. It's a good way to dip your toes in.
There are a lot of benefits to a cheap dedicated track bike if you have the garage space and a means of transport. The ability to customize the ergonomics, for instance. The seating on the rental CBR125/250s is a bit funky with the stock low pegs, and I found a bunch more confidence when I figured out that everything felt better when I could get my upper body more horizontal. So aftermarket rearsets and raising the seat height by several inches helped a lot when I eventually got my own track bike.
Buying a pre-built track bike is much cheaper than trying to build your own from a street bike. But if you find a smoking deal on a street bike and want to convert it yourself, budget for engine case protectors (couple hundred dollars), race fairings (1-1.5k), steel brake lines (ditch the ABS if it has it), and ergonomic stuff like rearsets and clip-ons, spare clip-on bars and footpegs. Stock suspension might be okay for a couple of seasons, but skip the engine/performance/power stuff for now.
I really like small bikes like the Ninja 300/400, R3, etc. They're much less physically demanding to ride so fitness won't be a limiting factor until you get really fast. They're cheaper to run (a tank of gas will last you all day, a front tire will last all season, a rear tire will last 5 or 6 days or more depending on your pace, even on the sportiest DOT hypersport tires you can find). A Ninja 300 took me all the way to red/expert group on a tight track like Mosport DDT.
Philosophically, I went with a bike that was cheap enough that it wouldn't hurt if I towed it straight to the wrecker in a dozen pieces at the end of the day. From an economic standpoint, I find that having a bike that is essentially disposable takes away a lot of the mental barriers about lean angle and low-siding. I've even learned my lesson about painting the fairings too nicely. It's better if it's at least a little bit cosmetically rough. I personally wouldn't worry about mileage at all.
Facebook is probably your best bet for finding track bikes, but I don't have Facebook so I've always just skimmed Kijiji. Set up a bunch of queries like "track bike" and "race bike" in addition to queries for whatever specific models you're looking for and check every couple of days. A bunch of them should be showing up soon as the season ends.
FYI, I picked up one of the newer Ninja 500s and converted it to a track bike over last winter, so my previous Ninja 300 is taking up space in the garage and I'd like to get rid of it. Dealing with the general public is one of my least favorite things ever, so I may just list it here instead of trying to survive Kijiji/etc. DM me if you're interested.
