Olympic Square is the first, and easiest track in Need For Speed: Underground. There aren't any super-hard turns, so keep your foot on the floor and try to keep your vehicle at or near its top speed.
The first turn that you reach after the starting line is the most difficult to get a clean line through; it's almost a 180, but without the simplicity of a pure hairpin. For the earlier portions of the game, you should be able to make it through cleanly by letting go of the gas as you enter, then flooring it as you exit. After you've modified your car with performance upgrades, however, you'll likely need to hit the brakes shortly before entering the turn to avoid ramming into the outside wall. A shortcut immediately follows the turn; cut across the oncoming traffic lanes to jet through.
After you take your next turn, you'll be faced with three paths, including both lanes of traffic and a tunnel. The tunnel offers no significant time benefit, and isn't a shortcut for the purposes of Style Points, but is generally preferable to travelling along the oncoming traffic lane, which has a blind jump that will occasionally see you flying head-first into a car you can't see. It's often difficult to cut back from your turn and make it into the correct lane of traffic for the direction you're heading, especially with a tricked-out car, but if you can reach it, then that's usually the best choice, since you'll be able to avoid traffic and will be in a good position for the next turn.
Said turn is a lengthy 90-degree angle that opens into a road with a tree-lined median. Try to slide into the outside lane here, erring on the side of caution, if necessary, by sticking close to the exterior barrier. You do not want to hit one of the trees here, as doing so will bring you down to zero miles per hour instantaneously.
The road splits in two beyond this little avenue. Try to take a right at the fork; this will prevent you from having to make a difficult turn where the road rejoins, and will give you a much better angle to enter the second shortcut of the track, which runs parallel to the dip in the road ahead. Hit the shortcut if possible, but beware of the cross-cutting traffic. Subsequent to that shortcut, you have an easy path to the finish line. As you head for home, be sure not to undercut any of your turns, as you will sometimes find yourself ramming into the corners of the track when you do this.
The reverse angle on Olympic Square results in a slightly tougher race. The important hotspots include the second shortcut (now the first), which is now a bit more difficult to hit properly as you round the corner towards it; you'll need to brake, or at least ease off the gas, to hit the entrance without sliding into the wall.
The fork in the road should, again, be taken with a right turn. It's tempting to go straight on these forks, but doing so results in a tougher turn a few second later, and as such isn't worth it.
The tunneled road offers the same set of choices as it did before, but here, it's more important to get to the outside before making the turn. The small drops in the road that you encountered on the straight course have turned into a couple of jumps that will prevent you from adjusting your position; before you go over them, you'll want to make sure that you're heading straight up the road and are not going to ram into any oncoming cars. Past those hurdles, take the second shortcut, round the corner (which is a bit tighter in the entrance than on the normal route) and you're home free.
The sewer section that leads off Atlantica will be a very familiar sight to you by the time you reach the end of the Underground mode; it pops up in quite a few of the Sprint and Circuit races. It's a simple section, excepting the pillar twins that split the path into three shortly after the beginning of the race. If you're runnnig alongside a car here, it goes without saying that you should attempt to ram them into one of blockades, but try not to be the rammee.
After you reach dry ground, you're going to face a hard left into a dirt section of the track. Taking this with a tricked-out car can be a pain; try not to go over 110 MPH or so as you head through, and tap the brake if you feel yourself sliding towards a wall. Subsequent to that little diversion is the well-used construction shortcut. Stick to the right wall here and take the jumps to cut a couple of seconds off your lap time. The left-hand section of this shortcut is to be avoided; it, too, offers a couple of jumps, but forces you to steer from right to left while you take them. Wall grindage usually follows, which is never a pretty sight.
The shortcut leads onto a familiar road with the obstacle-laden median. The subsequent track segment consists of four lanes of traffic, divided in the middle by numerous concrete barriers. The right lane of traffic is your friend here; the oncoming lanes are to be avoided at all costs.
The final turn into the sewers is preceded by a slow right. If you're in the right-hand lane, watch your speed here; the left-hand barrier has an abutment that will usually send you careening if you slide along the inside wall.
The backwards run towards the concrete barriers poses a quandary for drivers: the left-hand lane is a much easier turn after you begin the race, but leads into oncoming traffic, while the right-hand lane is a wider and sharper turn, but leads into the much safer traffic lane. It is possible to take the left-hand lane to make the first turn, then shift over to the right-hand lane after the turn is complete, but you'll have to dart through one of the gaps in the barrier shield, which is easier said than done, especially at 150 MPH. In the end, the decision will have to rely on the density of traffic: minimum or no traffic will make it easy to take the left-hand lane; any heavier, and you're greatly increasing your chances of running into a car.
The second difficult spot comes as you try to enter the construction shortcut; you'll need to brake, sometimes quite hard, as you round the corner towards it to reduce your speed enough so that you can manage the hard left into the left-hand shortcut. From there, the dirt shortcut poses another problem, but so long as you don't edge your car past about 120 or so, and brake appropriately, you should make it through without too many scratches.