Here are the potential issues with clinchers:
1. If you have a sudden deflation on a steep track, a tubular is hard enough to manage but a clincher is a good bit harder. Take care of your wheels, don't use worn tires, if it's an outdoor track be sure it's swept clean of glass (including the apron and infield circle), and don't touch tires with someone else, and you aren't likely to have a sudden deflation. If you can't control any of the above, you risk a sudden deflation and the almost inevitable fall that goes with it. Now I don't get too heated up about a fall, but a fall followed by three riders T-boning me at speed is something to think about.
2. You probably won't feel the same kind of stability on clinchers that you feel on tubulars until the pressure is a big higher than your usual road pressures -- say you do 140-150 psi on tubulars, you probably want to do 120-125 psi on clinchers (pressure depends on weight, track, your preferences, tire choice, etc. etc.). Some rims aren't all that happy with that much lateral pressure in the bead area. And some tire and rim combinations don't fit together as well as one might like, and high pressure can occasionally exacerbate a sudden failure -- it happens on the road (Vittoria Evo Corsa CX road clinchers had a pronounced problem for a while until Vittoria reduced the dimensions slightly) so you should anticipate it on the track.
3. Some clinchers create an egg-shaped profile for the tire. On the road it isn't bad, but as you go on and off the bankings at lower speeds (anything where you are more upright), you can change the contact point on your tire and feel a bit less stable. This is easily avoided with a fairly round-profile clincher (like a Veloflex Pave).
4. It's hard to get clinchers down to the same weight as tubulars, and if you want to use higher pressures it's hard to get them up to equivalent pressures.
5. The high-end track wheels are pretty much only available in tubular. Same for the best track tires. If you aren't aspiring to be a high-end competitor, this may not matter to you.
6. As you progress, you are more likely to want to be on tubulars. When you're good enough that they make a difference, they're really nice to have. You might as well learn how to use them properly from the beginning.
What do clinchers have going for them?
1. Ease of use -- that's the primary reason they're used on rental/class bikes at tracks. I used to have to glue on all the tubulars for track training programs, and believe me, you get really tired of it. And if I don't like the look of a tubular wheel, I have to allow a couple days or more to re-glue it, while I can be on a clincher wheel in ten minutes. I ride tubulars myself on the track, but I have a couple pairs of track wheels built with clinchers that I'll throw on if I'm in a jam for a functional wheelset (or if I want to loan wheels to someone and don't want them flatting my expensive nice track tubulars).
2. If you ride on the road as well, your proclivity for flats may make clinchers more practical if you use the same wheels for both. Someone riding fairly serious track will often have a pure training pair of wheels (or more) and then pure racing wheels and they will fairly often be tubulars. When you're at a track event and trading equipment around, you get to borrow better wheels if you have better wheels to offer when you need to lend them.
3. If you can't stand gluing tires, or can't seem to do it right, stick to clinchers.
4. Costwise, you probably come out a bit ahead on clinchers. I say "a bit" because you can get amazing mileage out of a good but inexpensive tubular like a Conti Sprinter. Frankly, if you pick appropriate tubular tires, you can probably make the costs work out even.
5. You find all kinds of sub-communities at the track. Some think tubulars are essential; others think that clinchers are the only way to go. Depending on the team you're on, the people you like to ride with, the events you want to do, etc., clinchers may fit with your riding.
Frankly, don't ask on this forum. Instead, go to the track where you'll be riding and see what they're using. I've never seen a track where people weren't friendly about answering questions for newcomers, or even letting you try different wheels to see how they do for you. Local track conditions and preferences predominate, and what the crowd at ADT likes and what the crowd at Dick Lane does may be very different. To start, you can ride just about anything, then decide how good you think you want to be and how much you want to spend. Borrow some wheels on training nights and see how you like them yourself, and listen to local opinions. In the end, spending money on racing is more fruitful than spending it on bikes or wheels. If you want to be riding nice wheels as well, decide what you want to get nicer wheels for (training, power events, pursuit, massed start, etc.), and then pick wheels appropriate for that application. For better wheels, the recommendations become different for different events.