Learn the Track
This is probably the hardest, but most effective method for speeding up. With a good knowledge of the track you can pick up 2-3 seconds a lap (and that's a pretty conservative estimate!). There is no easy way to learn the tracks except to drive round them a lot! Look out for landmarks, such as the dome at Mexico, the bridges at Silverstone and the buildings at Monaco. (Of course, if you have a slow machine, it's better to turn detail off and learn the courses without landmarks, so you can get a higher frame rate.) Once you know what corners are coming up you can anticipate the way that the opposition are going to move and which side of the course to position yourself on. Of course with the Ideal Line on it becomes a lot easier to tell, but on the higher levels you lose this option, and the ideal line isn't always ideal anyway.
OK, so you know where the corners are, now how about slowing for them. The Auto-Braking feature is very poor, and you can usually travel another 50-100 yards before you have to apply the brakes, this is the main way for passing the other cars and making up a lot of time. The computer controlled cars always brake very early so it is east to shoot past them, there is no excuse for not passing 3 or 4 cars at every 1st or 2nd gear corner.
By late-breaking into all corners whilst on a clear lap you can easily smash the lap record, the only problem with this technique is that your tyres get worn quickly and you may find yourself having to make an extra pit-stop.
A good start can gain you plenty of places, or keep you out of the usual pile up at the first corner. If you have too many revs as the lights go green you will sit stationary whilst the wheels spin, and if you have too few revs the car will move away too slowly. There is quite a large rev-band in the middle where you can make a very good start.
My personal technique is to hold the car at full revs in first until the first lights come on, and then let go of the clutch (gear change button), by the time the lights go green the revs will have fallen enough for you to make a pretty good start.NB - this does not apply in the wet, when two many revs causes massive amounts of wheel spin.
My technique is similar, but I let the revs start dropping after counting to 5 (about 7 to 8 seconds). This is slightly before the red lights come on, about 1-2 seconds. That way, the revs are a little lower when the lights go to green, you get less wheelspin, and a quicker start.
If you aren't on pole, you should be aware that when the wheels stop spinning and bite properly, you'll get a sudden burst of acceleration which can easily put you into the back of anyone in front.