The carry rig, along with stuff, will be cost almost as much as the gun. I recommend the Comp-Tac kydex reinforced belt. It's an excellent belt that holds up the gun and ammo nicely. Add in a good horsehide IWB holster from Milt Sparks, Brommeland, or similar along with two matching single magazine holders and you'll be good to go. Consider buying a DeSantis Cozy Partner in addition to the other holster. The Cozy Partner is an excellent, and comfy, holster with more forward cant than most other holsters. The "problem" with it is that the holster lacks a metal reinforced mouth and will collapse if it gets sweaty. This is fine for daily carry, but not so good for training classes.
The Glock 19 is a great gun and is easy to conceal, especially in a DeSantis Cozy Partner. Consider the overall height as a very good rule of thumb for concealed carry. There are quite a few equivalent guns. You should be able to find a used Glock 19 for around $400.
One gun to consider is the Browning Hi-Power MK III. You get 13+1 capacity (use only Browning magazines; others seem to be a crap-shoot) in a very easy to carry and shoot package. Add some nice cocobolo grips from Herrett's, some night sights, bevel the magazine well, tune the trigger and you'll have a very nice gun.
An even more concealable pistol is the Kahr K9. It is a single stack gun in 9x19mm. Sometimes Kahr quality can be hit or miss. Mine had reliability problems, but other people I know have had excellent results with their K9s.
The Glock 26 or S&W M&P 9 Compact are great little carry guns. I have had excellent results and reliability with the Glock 26, but only minimal exposure to the M&P 9 Compact.
The Springfield Armory EMP in 9x19mm is another excellent gun. It is the same height as the Glock 19, but the slide is a bit shorter. It comes ready out of the box with night sights, ambidextrous safety, beveled magazine well and nice wood grips. It is very accurate and mine had no trouble digesting steel cased Tula ammo as well as JHP ammo. You should be able to find one used for around $900, but it may exceed your budget. I think it's worth saving for.
Another option would be to buy a rig for your Model 10. Add a front night sight and get the trigger tuned. Buy some boot style grips and you're done. You will be a very, very good shooter if you can master the double action revolver trigger at speed. Buy a reloading setup for 38 Special, several thousand rounds of 158 grain 38 Special +P jacketed semi-wadcutter hollow point ammunition and practice frequently.
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