I just got my first pistol , M&P 9 . I put about 300 rnds through on Sunday without a failure. I'm glad I didn't start with the .22. Also it has the trigger upgrade but I can't comment anything on that as I've never used a stock trigger. The only complaint I have , are the sights, I believe they're more suited to tactical shooting more so than precision. ( white dot sights)
You're right about the sights on the M&P. They are designed for quick target acquisition, not slow-firing accuracy. The M&P, as I understand, actually stands for "Military and Police" as that was the idea/intended market behind its design. Glocks are the same, so is my Ruger SR9.
While it will never be a "target" pistol likely, don't be afraid to adjust the sights so your groups are going where you want them. If you get some silhouette and/or tactical targets you'll see your shots in a different light.... bulls eye targets will make you feel like the worst shot ever, lol.
As for the OP, with that budget you have lots of very quality choices. Personally I'd go 9mm, you can learn to shoot decent with that round, lots of people have and do. High quality guns like Glocks, S&W, Ruger, Sig, Beretta's will last generations if they are treated right so don't be afraid to check out the used market, you could score an awesome price and have lots of money left over for whatever else you might want to buy (cleaning supplies, ammo, holster, belt, extra mags, targets, sight upgrades, etc, etc.).