There's no quick way to get good at them. It takes time to build up the technical skill, a concept that's not any different from playing games like Dark Souls or FIFA. If you play it long enough you'll eventually get better.

Getting good is mainly about practice,practice, practice. Learning the track and the car.
Some corners you don't need to brake for and can just coast through by simply letting go of the gas pedal or downshift, other corners of-course require braking. This is why you need to practice, to learn where to do what.

Sound is very important, you need to hear when it's time to shift up or down, which again comes down to experience. The best way to learn the mechanics is to have as few assists on as possible, choose manual shifting for example. There will be lots of mistakes to begin with but that is unavoidable because it's all part of the learning process.

So initially it's all about trial and error, but eventually it becomes more about experience. As you get better you also learn faster.

As for games, it really depends on what kind of game you are interested in.

There's the sim-arcade hybrids, games that have elements of PC sim-racing but are much more forgiving:
F1 by Codemasters, Dirt 1-3, the WRC rally games, Forza and Gran Turismo, Shift 1-2, Grid Autosport etc.
These usually give you sim-racing options such as manual shifting and car setting adjustments.

My favourites among these if you aim for more sim-like racing would be Dirt 1, F1 2011, WRC 4, Forza 3, GT 6.
I find that the AI in games like Shift, Grid and the later Forza games to be unrealistically aggressive.
These are just my personal preferences mind you.

The more "hardcore" racing sims, which are more demanding to learn, include:
Dirt Rally, Assetto Corsa, the SBK superbike series, the Race 07 collection, the RaceRoomExperience game and it's many expansions (DTM, ADAC, WTCC ... ), the Reiza Studios games (Game Stock Car 2012, Stock Car Extreme, Formula Truck, Automobilista), rFactor 1-2 and iRacing.

If you don't want to use Steam, the Reiza Studio games can be bought directly from them, you get an activation code which can be renewed simply by dropping them an email. I have bought three of their games in this way and have not had any problems. They have demos you can download..

iRacing is subscription based, just like an mmo, which is why I haven't stuck with it. Great game though.

The core game in rFactor 1-2 come with plenty of stuff to be worth the price in my opinion, but the bulk of the content - the modded stuff - is mainly accessed through subscription fees. I think you can still buy the rFactor games directly from ISI in case you don't want to use Steam. They also have demos you can download and try.

The core of RaceRoomExperience is free, but you don't get much free stuff, instead you buy individual cars and tracks, or buy packages of cars and tracks, or buy the so-called "Experience" packages which come with all the cars and tracks for a specific racing championship (DTM 2013 Experience, ADAC 2014 Experience ...)
There's no subscription fee but you can't play it offline. There's no demo but the core game is free, so ...

I mainly play in single-player mode, and apart from iRacing they all work really well for this. For online multiplayer iRacing is the best, it's what it was designed for.

Even the more "hardcore" PC racing sims have an arcade-sim game mode for people who want something more forgiving but still want to experience the rawness of these games. So don't be intimidated by the "hardcore" PC sims, everyone can play play them, not just an elitist few. Contrary to popular dogma they all work really well with a gamepad, or even keyboard.

There's some really good forums around for you too look for more information, for example how to set your car or driving wheel: , [url=http://forum.reizastudios.com/]Reiza, and [url=http://www.isrtv.com/forums/]InsideSimRacing.

Most games of sim-like nature have some kind of forum or forum section dedicated to it.