If you're looking for quick cash, you can't do any better than a Drag race. These take place on a long, straight stretch of road, where you'll have to scorch off the starting line and beat the other three cars to the finish.
Your car controls much differently in a Drag race than it will in a regular challenge. For one thing, you won't be able to use an automatic transmission; manuals are mandatory. Second, steering is automatic, except for lane changes, which are done automatically when you move to the left or right. This may sound like it would result in a difficult race, but in fact, Drag challenges are among the most fun races in Need For Speed: Underground.
Although the first car across the finish line is the victor, Drag races are really won at the starting line. Before you take off, you'll notice a tachometer along the left side of your screen; the needle here changes from blue, to green, to red, corresponding to the comparative readiness of your engine to upshift. While you're waiting for the race to start, you'll want to attempt to lightly press on your accelerator so that the needle stays green; this will give you a Perfect Shift to start the race and should let you jump out ahead of your competitors. It's difficult to maintain a green needle, though, especially on the Gamecube or PS2 controllers; if you're having a tough time with it, keep in mind that it's far better to shift early than late.
The same holds true after you begin the race. Immediately after a race begins, your car will be ready to shift into second gear, so watch the needle as it falls and rises, and try to time your button press along with the green needle. To aid your shifting, a blue dot will appear in the top-center of your screen to warn you when the perfect shifting opportunity is approaching; it will turn green during your window of opportunity. Shifting in the blue can either result in an Early Shift, which is bad, or a Good Shift, which is, well, good. Shifting in the red, however, will result in an Over Rev, which will take a bit longer to recover from than an under-shift. It's incredibly difficult to overtake a car after they pull away from you; if they're a few car lengths ahead, you can attempt to draft them, then pass, but if they're beyond drafting range, you usually will be forced to restart the race.
Early Drag races are simple affairs, but as you progress through the game, you can expect to see a few complications arise here and there, such as lanes suddenly ending, oncoming traffic, cars that pass across the track at intersections, ramps, and so forth; you'll need to stay on your toes if you don't want to total your car. (Although it must be said that the slow-motion replays of spectacular crashes more than make up for having to restart the race.) In addition to the normal style bonus for winning the race, you'll also be able to rack up points for near-misses, hang time, head starts, and drafting. Drafting becomes especially important as you begin to race against better cars; if someone gets a jump on you, shift over behind them and attempt to get a speed boost. Then pay them back by coming alongside them and ramming them into an oncoming car.
Unlocking the turbo and nitrous upgrades for your car will also increase the difficulty of Drags, or, at least, increase the amount of technical skill required to get a good jump off of the starting line and obtain Perfect Shifts. You'll notice that the red zone on your tachometer will shrink after installing a turbo kit; this will reduce the Perfect Shift zone required to get a good jump, so watch the needle carefully before you zoom off. Turbo kits also make it nearly impossible to get a Perfect Shift into second gear, due to the way the needle jumps after shifting into first, but a Good Shift shouldn't hurt you overmuch.
Nitrous systems can also make your life more difficult. Keep in mind that after you install a part onto your car, all of the other cars that you're racing against will be equipped with the same thing, and they'll generally be more proficient in their use of nitrous than you will be. You'll likely not want to hit your NOS before you reach third gear; doing so before that shift will usually force your needle into the red, and lock you into an Over Rev shift into fourth. You can use nitrous in bursts between your early gears, to get into position for a Perfect Shift, but again, this puts you at risk of an Over Rev if you don't time your burst properly. Of course, after you hit fourth or fifth gear, you'll be able to drain your NOS for a huge speed gain; this can help you catch up to a car that's pulled away, or just extend your lead a bit. Most of the computer racers will drain their nitrous early, so don't panic too much if they jump out ahead; just get behind them, hit your own nitrous, and blast past them after drafting. Just make sure not to push your engine too far with nitrous; blowing an engine is as bad as totaling.
If you're having a hard time getting a jump off the line, try going into your garage and reverting back to a stock turbo kit. Doing so will force your competitors to remove their turbo kits, as well, but will also make it much, much easier to hit early Perfect Shifts. Revving up before a race will still be difficult, but losing the turbocharger will eliminate your tachometer's proclivity to jump before shifting to second gear, and will give you a much bigger window in which to get Good Shifts and Perfect Shifts. If your competitors keep bursting out in front of you, you can attempt to take off your nitrous system as well. Some challenges, however, will detect your car's capabilities and force you to install these parts if you've taken them off.
Although blocking is important in races, it's especially vital in drag challenges, where you can push your opponents into obstacles or cars, or simply prevent them from passing you until you hit the finish line. Use your rear-view mirror to check on any cars that are drafting you, and try to get out of their way so that they can't gain a speed boost; if they try to race by, jam over into their lane. If you actually hit an opposing racer while changing lanes, you'll either be repelled back into your lane, or they'll be forced over; your chances of getting a good result depend on the position of your cars. If you're slightly ahead of their front end, you'll likely win; if you're hitting their rear, you won't accomplish anything useful.
In Underground mode, you'll occasionally come across Time Trial challenges, which are effectively solo Sprints. Instead of racing against other cars, you'll need to make your way from the starting line to the finish line before the pre-set time limit is reached; the prize for these kind of races is usually a unique upgrade to your car that cannot be bought in the garage.