I suggest a Keltec, probably the P32. It is a double-action only with a heavy-long trigger so that the trigger is less likely to be inadvertently pressed than a single-action like the Colt and therefore there is no need for a safety to take-off and put back on in the manual of arms.
Keltec also makes the larger P3AT that shoots a larger .380 ACP cartridge. The Ruger LCP is a copy of this gun -- and the Ruger is by far the #1 selling "pocket pistol" and has been the #1 selling gun (on and off) for years. I suggest the P32 because it's even smaller and you can feel totally confident in concealing it without significant changes to your wardrobe or uneasiness about it being exposed. It should be in a pocket holster or an IWB holster.
It's good that you're training, and I would suggest training with a full-size gun (Glock, Sig, Beretta, Colt, whatever) for the first 60 hours or so. By training, we're talking about instructor-led classes. Don't take a P32, LCP, Model 36 or any pocket pistol to classes if your fundamentals aren't sound.
Now some might question, why would I say to carry it then? That's a good question. But you really have three other choices. Get a pocket pistol or little snubbie and try to gain skills with a crappy gun to start with; Don't carry anything until you've gained skills and then transitioned those skills to a pocket pistol; or try to carry a full-size gun.
The smartest thing to do is to carry a full-size gun, but it's apparent that you won't do it. I'm not going to blame you for that. You have your reasons. I will advise you though, not to try to gain skills with a crap gun before you even have them with a good gun.
So if you follow my advice to use a good gun to work on skills, you can either not carry until those are developed and then transitioned to a pocket pistol, or you can get the easiest gun to carry (P32) and hope you aren't pressed into using it until your skill level comes up. Following this strategy with a Model 36 or a larger pocket pistol like the Colt doesn't make sense. Those guns are both hard to carry and hard to shoot. Worst of both worlds.
Once you've been training with a good full-size gun, and habitually carrying a gun, you will be forming some good ideas of what direction you'll want to go from there. Maybe you'll feel like you could easily carry a lot more gun. Or you'll start shooting the P32 more often and realizing what a PoS it is or is not -- depending on how you get along with it. When you can, rent more guns according to your accumulating experience -- maybe you train with a Glock 17 and decide you'd like to try a rental Glock 43, or you train with P320 and want to try a P365, or you train with a 1911, but convince yourself you don't want to carry single-action. You like the P32's DAO, but want something bigger. Rent a Kahr in 9mm. You'll get there, but there are no good shortcuts.