Most folks don't need any changes done to the LCP.
My LCP, even after smoothing up obvious friction points (polishing the feedramp, smoothing out various things related to the slide, etc.), was a bit cranky about fully feeding the first round from a fresh magazine. That's even with rather vigorous "slingshotting" of the slide. Installing a Wolff 11# recoil spring set in place of the factory 9# spring set made all the necessary difference. I could probably go back to the 9# spring set after having shot the pistol a bunch now.
The trigger on the LCP is a little hard to pull compared to other pistols and, of course, it has a lot of travel. Bear in mind that the trigger design is the ONLY "safety" on the LCP, so I would strongly advise against replacing it with some of the fancy adjustable trigger kits that are being sold today.
Some folks have the sights replaced with taller and more visible sights. Each to his own, but I don't even use the sights on the LCP when doing controlled "point shooting" which, in my opinion, is the way the LCP was designed to be used.
You would probably do well to put the finger extension on the magazine and get a couple more magazines with the finger extensions on them. That can give you more control over the pistol, but it does make it slightly harder to conceal the LCP.
Make SURE that you get and use a decent holster (or several different ones like some folks here do) that protects the trigger from inadvertently pulling it when re-holstering the LCP.
Get lots of ammo and practice combat-style shooting. That means rapidly pulling the trigger while keeping the pistol pointed at the target. Doing a lot of dry firing also goes a long way toward improving your performance. Just make absolutely sure that you remove all live ammo from the room and check several times that it's out of the room before starting a dry-fire session.