Can't resist a good handgun discussion, so here (after a busy-week-protracted-absence-from-the-forum) is where MtnMan gets to chime in.
There is actually a Glock-based solution to both the size and caliber solution - K-Bob feel free to correct me here as it has been a few years, but I carried a Glock 36? in .45 ACP which actually fit in a pocket holster. Not quite as easy as the Smith 637, but still fit in the pocket and I like the .45.
(By the way, I've never been overly bothered by the .45's recoil, it's actually way less than an airweight with +P rounds in .38)
Having said that, I have to also admit that I traded the Glock for a Taurus Millenium Pro (also in .45) because it gave me 4 extra rounds to carry, although it didn't quite fit in my pocket (a Fobus on the right hip was perfect, though). This was in response to a perceived increase in threat as I started making frequent innercity trips adjacent to drug-infested neighborhoods - wanted a few extra rounds in case I happend to get caught in a street war going in or out. I have since traded it as well, and am fairly happy with the Keltec P-11 in 9mm. Not my favorite carry piece (see below), but I have shot it often enough to be comfortable with its handling and aiming characteristics, and 10 rounds of 9mm is a decent load for a lowered threat level.
I have carried for many years and have tried many models, and my favorite was probably fastmover's suggestion, the 637 airweight was in my pocket for 3 of those years and I reallly miss it. The revolver carries very sweetly, shoots like a dream, and with careful shot placement has plenty of punch. If I should locate another at a decent price, the Keltec will be replaced rather instantly.
For those of you about to quote the Phillipine disaster as a negative, remember that we had pretty crappy bullets for the .38 at the turn of the century (and indeed up to the 60's). A snubby fitted with today's loads is a potent protector, even without going +P, and a revolver avoids many reliability issues right out of the box. A quick clip, or even a speedloader, adds to the capability.
So, though it would appear that I am all over the place in my carry selection, it is more that I HAVE been all over the place, and if I was going to start all over, as the original poster is, I would buy a S&W 637 or possibly a 642(hammerless) and never look back. Your mileage may vary, but the .38 snubby was specifically designed for this purpose, and done quite well.
Are there better calibers? Yes. Are there smaller, more concealable guns? Yes. Are there situations where I may need/want more power or capacity? Yes. But I stand by my choice for the newcomer to concealed carry. You have a lot to think about, consider, and learn in regards to situational awareness, prevention tactics, exit strategies, and the whole "should I" mindset - caliber and capacity should not be primary concerns at the outset. The snubby in .38 is more than enough protection platform while you ponder these other issues.
Having seen Unca's suggestion a second ago, if you want the .357 instead, that's fine. You can shoot .357 or .38 in it, but beware that recoil with the magnum is best described by the word "punishing" and will slow down follow up shots - I had a somewhat heavier gun in .357 which helped some with the recoil, but it was a real pain for pocket carry due to the additional weight (a german Windicator I believe, about 39 ounces, and recoil was still "stiff" in my opinion). I probably wouldn't suggest starting out with the full bore loads, but Walt's choice would allow you to go there later.