Concealed Carry permit or license to carry firearms, as it is in PA, requirements vary by state and apparently even by county (from what I've heard from other PA license owers). In PA, all you have to do is get a form from the county sheriff, fill it out (requires 2 or 3 non-family, same county references and a head and shoulders photo of yourself), and then hand it back in to the sheriff. They then do a background check which takes a few weeks. If you pass, they let you know and you go pick up your license. I believe it was $19 for a laminated license in my county.
As for which gun, that is a very personal choice. A gun that fits my hands well, feels comfortable, and is of a caliber I can handle may be the complete opposite for you. As others have said, if you can go to a range that rents them, that is the best way to test them out.
Another thing you'll want to think of is ammo cost. .22 is the cheapest at about $13 for 550 rounds (used to be less than $10) but also the least powerful. 9mm would be the 2nd cheapest. I can get a case (500 rounds) of 9mm for $80 plus tax right now at Dick's Sporting Goods (when they have it in stock). The same amount of ammo in .45 or .380 is $150 + tax - almost double the cost of 9mm. Not sure about .40 but I'd bet its close to the same as .45 costs.
If you buy a single box of the same ammo at dick's, it about $10 or $11 a box for 9mm and about $17 or $18 a box for the .380 and .45, so buying in bulk or cases at a time is the best bet.
I have 5 pistols - CZ P-01 9mm, Taurus PT99 9mm, Springfield Mil-Spec 1911 .45, Bersa Thunder .380, Ruger Single Six .22 revolver.