First off, congrats on finding a new passion.

Secondly, you are definitely not too old. Nowhere near it. You've got at least another 30-40 years of good racin' left in you

What track is your local track? If you can't make a beginner class there, maybe another track within a short drive will have one. These classes make things a lot easier and help new riders get the jitters out. Most velodromes require such a class, or some sort of proof of having completed one at another track, before racing. The stuff isn't hard. It's just necessary to keep everyone on the same page when out on the track.

As far as getting a bike goes, it depends. It depends on your financial situation and it depends on how well you know yourself. It's sort of like being interested in playing the guitar. Are you interested, or are you REALLY interested and sort of know that it's gonna take some work? If you think you are going to stick with it for a while, then work towards buying a decent bike (or guitar) once. If you aren't sure then maybe borrow/rent one for a while till you have more info about whether you want to go further with the sport.