Though every player is going to have a different style that works best for him or her, there still seem to be some general strategies that will carry you further than others.
The first thing to remember is that tire upgrades are much more important than you would expect, perhaps more important in some cases than motor modifications. Early on, your first upgrade shouldn't be the red motor (which mostly is useless). Instead, go for the Skinny Tires and watch how much time you cut off most corners. From there, your next upgrade should likely be the silver motor, then the nobblies. In this way, you can keep your turns efficient and your motor in decent shape. Invest on the smaller motors in between and you're just wasting your money.
With good tires, you should really get the most out of corners. Even if your opponents can beat you in straight stretches, sharp turns can work to your advantage. This is particularly true of the elbows that switch back on themselves repeatedly. Going into the first bend, align your car toward the center, straight ahead. A diagonal line will zip you through the corner while your opponents are busy with a bunch of time-consuming turns.
Another thing you should note is that the letters are very important to your success. When possible, try to be the first to a particular vehicle upgrade. You'll get a boost in speed that the early models don't have. With that speed, you can repeatedly finish in first place, possible for an easy track or three, before the others catch up. The difference in cash can get you another upgrade while your opponents are still struggling.
Finishing first isn't necessary in the game (you only have to finish third or better to avoid losing a continue), but it sure doesn't hurt. You'll get more points, and there are several points in the game where your points are tallied for potential cash bonuses of as much as $3,000. That's free money you definitely want, so get as many first-place finishes early on as possible.
As far as weapons go, they don't seem to be particularly important in the single-player game. Buckshot is fun in multi-player when you want to tick off your friend, but it's not practical in the one-player game. If you do choose to buy a weapon, the freeze shot is probably your best bet, as it can save your butt if you're racing someone just a little bit faster than you down the home stretch and it's looking like you might not rank without some cheap shots.
Then there are nitrous to consider. When you pick them up, they only work for the race in question, so go ahead and use them to your best advantage to finish ahead of the pack. Note that if you're in the lead for a race littered with nitrous, this also makes it easier to grab the upgrades. Start in the lead early and you shouldn't have much trouble maintaining that position, in other words. Also, the nitro boosts are particularly useful at the base of hills. They can carry you past the competition while you're airborne, and you might even bounce over puddles, bombs, or other obstacles that otherwise would slow you to a crawl.
Remember that when you've finished the first 24 tracks, the game will just pick 8 more tracks out of its favorites to toss your way, but they'll have modifications to keep you busy. For example, one might have more nitro pick-ups, another more bombs. Don't assume that just because you've beat them before you'll have an easy time the second trip through. Note that while you'll continue through the league in the first 24 tracks even if you come in fourth, that's not true of the final eight.
This actually works out near the end to your advantage. One of the levels starts with a 1-up icon just behind the finish line. Turn around and snag it right away, then finish in fourth place. You'll get a cash bonus, and continue where you were. Since your continue was in essence a freebie, it doesn't really cost you anything to do so. It's a great way to build up cash for final vehicle upgrades, or for continues or whatever else you want.
Finally, if you'd like to get a quick start on a race, wait to press the 'B' button until the split second the final light flashes at the starting line. Immediately press the button and your vehicle will get the equivalent of a nitro boost that can allow you to take the lead early.
Of course, none of the above strategies will take you so far as memorization of the tracks themselves. Get to know their hazards well and you'll go far, perhaps to the end of the game.