Read this ten times and then read it again.
http://en.lfsmanual.net/wiki/Advanced_driving_guide
Read anything and everything you can about your favorite cars and your favorite tracks no matter how basic the information might seem. Soak in all the information and try to apply that information in baby steps on the track. Don't overwhelm yourself.
The best advice I can give is to pick your favorite vehicle (the one you feel the most competent/comfortable with) and practice, practice, practice (mostly off-line in the beginning); hot-lapping. Stick with that car and learn its strengths and weaknesses. To ease any frustration, pick your favorite commonly raced online track and start logging some seat time. Have fun.
Then do a few single-player races on various difficulty levels. The AI is pretty lacking, but just follow them and observe. See if they are taking some faster lines. You might learn a thing or two.
Go back and hot-lap a little bit and see if your times improve. I guarantee they will. (At least a little bit.)
Once you feel comfortable, find a server that is running the track you like and enter a race. Treat the session as an observation session more than a true race. You want to compete here, but don't necessarily "race to win". You won't. You're still gonna get spanked. Have fun.
Online racing will be a totally different experience from hot-lapping as you probably already know. Now you have to combine fast lap times with consistency while at the same time maintaining situational awareness and adjusting your racing lines and braking distances. Watch the lines the human racers take. They will be a little different from the lines you've probably been taking. You're still learning here. Have fun.
Save the replays from your online races and watch them from the "in car" views of the faster drivers. Compare the fasters drivers styles to yours and practice, adjust, and improve. (Make sure to "Display Pedals" in the options list so you can see when a driver brakes, applies full throttle, etc.) Learn when they are braking, going full throttle, coasting, up-shifting, down-shifting, etc. Don't copy them. They might have a different setup, driving style, or controller configuration. Just learn from them and try to get faster and, most importantly, more consistent. I am constantly learning something new just by observing replays and chatting in practice sessions. If you aren't in the middle of a race, politely ask one of the faster drivers something that might benefit you. You might learn that they are taking a certain turn in the next gear up or down than what you were trying to run and bingo...you just learned something new. I hardly ever enter a server and find people unwilling to help. That's what seperates LFS from a lot of other "racing sims". Trust me here.
Once you start to get comfortable, you might want to start tinkering with your setup. Setups help, but driving skill and consistency is more important. You can always politely ask someone to send you their setup, or a player might offer to send you their "set". You might want to try it, but take it with a grain of salt. They might have a totally different driving style or might have their setup tuned because they are using an X-Box controller or a mouse and keyboard. You might end up with a jewel, but often times it is a big disappointment. Buyer beware. I often try to make my own, but if I am lazy or I don't drive a particular car much, I might take the best traits from several sets and combine them into one that suits me. I am terrible at setting gear ratios, so normally that's the only part of a setup I usually want. Many setups are track specific and the shift points are optimized for the corners and straights.
You need to sit down and narrow down the reasons you are not making the gains you want and start improving in those areas one step at a time. When you get frustrated, shut it down and go do something else. Come back when you are ready and settle in.
The day will eventually come when you finally "get into the grove" and everything just clicks. That's when the fun begins. LFS has a steep learning curve and you will find that you get what you put into it.
Above all else...have fun.
P.S. Once you are really confident with your favorite car, start branching out and trying different cars and different tracks. You'll find everything getting easier because your skill level has improved. Remember, there are usually 6-20+ cars on the grid in any particular public server.
There can only be one winner. Nobody is going to remember that such and such won such and such race on such and such server Thursday morning at 08:54 am GMT. Race within your limits. Enjoy the atmosphere of the race and treat every lap as a learning experience. Don't take that the wrong way. Never stop being competitive. Once you really start to improve...Race to Win! Keep practicing, keep pushing to improve, and you'll get there. Don't let these guys bash you on your grammar. Beat 'em behind the wheel.