I don't know if the old thread will be transported over to this forum or not, although I think it probably will be. In the mean time, I thought I'd address a few new points.
Ok, you've chosen a handgun, have gotten some training, and are wondering what's next.
Almost any factory handgun can be improved upon. Here are a few key common upgrades.
1) Trigger job
Almost any factory gun can be improved by a qualified gunsmith working the trigger. Factory pieces are machine made and will invariably have some rough surfaces, sloppy mating, etc. that just goes with mass production done by machines. A gunsmith can hand fit and polish your internal trigger mechanisms, which will result in a smoother and better feeling trigger. Almost any gun will benefit from this, to a greater or lesser degree.
In addition the smith can "tune" a trigger more to your liking. Trigger weights can be adjusted up or down, over travel can be adjusted or eliminated (the amount of distance the trigger can still travel backward after firing a shot) as can reset distance (the amount of distance the trigger must move back forward before being ready to fire again).
GENERALLY speaking, a lighter trigger is easier to shoot. Remember the intended use for the gun. A 1.5 lb trigger is fine on a target gun, but too light for a defensive carry gun, as it would be too easy to accidentally discharge. A 4.0 lb trigger is about as light as you probably want to go on a carry gun.
Less over travel is always better. A shorter reset is always better.
Note trigger weight can only be adjusted so much, especially in a revolver where part of the weight comes from the hammer spring. If the hammer spring is too light, it won't cause the hammer to fall hard enough to consistently ignite the ammunition's primer, resulting in a failure to fire.
Glock triggers can be VASTLY improved with a Ghost connector and a competent smith's polishing. M&P pistols can also be vastly improved by an Apex kit combined with polishing.
2) Sights
The quality of factory sights varies widely. Some are plastic with painted in dots that will fall out under heavy use. Others are steel with tritium vial inserts. Crap sights should obviously be replaced, but sometimes even high quality sights just aren't quite right for an individual shooter. You may prefer a wider or narrower rear sight gap, you may prefer more contrast or different colors, etc.
I recently replaced my Sig P220's factory night sights because during the day the milkiness of the front dot made it slower for me to acquire. I went with Trijicon HD (orange), as pictured here:
I find the orange dot much faster to acquire, especially with no competing colors on the rear "u" channel sight. At night there are three green dots, like standard night sights. This simple change has improved the speed and accuracy of my follow up shots noticeably.
3) Grips
Not all handguns have replaceable grips. Especially with earlier polymer pistols, you can put a sleeve over it but there's no way to make it smaller or really change the shape to better fit your hand. Newer polymers tend to have replaceable back straps so you can at least change the size up or down without resorting to a sleeve. Revolvers really shine in grip selection, though, as there doesn't have to be room for a box magazine. You can go way oversized for target pistols down to really thin stocks for better concealment to anything in between to better fit your hands and goals.