The game takes place on a number of different planets. The planets you can access depend on the difficulty level; on Rookie you can only race on the first three. On Veteran you can reach the fifth (Nho). On Warrior level you can compete on all six planets.
Each time you reach a new planet you start racing in Division B. To reach the harder Division A you'll need to earn points by placing in one of the top three positions.
Once you have enough points in Division A you can leave the current planet and move onto the next one.
You can also earn money during races by blowing up competitors ($1000) or by driving over the money on the track ($1000 each). If you do really well you can earn a $5000 bonus by lapping the driver who is in last place and then finishing in first place.
Each race has four contestants (including yourself). The twins Rip and Shred follow you from planet to planet; Rip drives a gray car and Shred a brown one, and they get newer vehicles during the course of the game. The fourth competitor is a local racer and is usually more of a challenge than Rip and Shred.
This is where you start the game. You can't race without a car but you only have $20,000, which means you must buy the Dirt Devil or Marauder. As discussed on the Cars page, the Marauder is usually the better deal.
At this point, you may wish to avoid upgrading your vehicle and save the cash for the Air Blade. You will probably not make it through Division B on your first time through (on the higher difficulty settings), but that often will not change whether you upgrade your vehicle or not. You should be able to buy an Air Blade on your second time through Division B, or if not, your third time. Once you do, you'll have no trouble getting through the remaining races on Chem VI, even in division A, with no further upgrades. But do upgrade your Air Blade when you can afford it until Drakonis (where you'll upgrade a bit more, then save up for the Battle Trak). You'll want to buy Rhino Skin armor, some plasma rifle shots, and possibly a new engine, but you won't really need the engine upgrade until Drakonis. If your character has no cornering, upgrade your tires, too.
An alternative is to skip the Air Blade and keep upgrading the Marauder, but this is a very expensive proposition as a heavily upgraded Marauder will still underperform compared to an Air Blade. On the whole, buying an Air Blade and upgrading that instead will be both easier and cheaper.
You will probably get more than the required points before you complete all the races in Division B (if not, it's a near certainty in Division A). This gives you the option to go to Captain Braddock to go to the next planet. But don't do it! You have nothing to lose and free money to gain, which you can use to upgrade your car, and that's always useful no matter how sick you get of the tracks here.
Remember that the Battle Trak costs $110,000, and that's how much you will want at minimum when you finish Division A, since you'll want the Battle Trak on Bogmire, and you need it if you have no cornering. If you like, upgrade as little as possible and only buy upgrades when it's apparent you'll need to in order to win enough races to get the points you need.
Buy the Battle Trak immediately! It should be enough to get you through your first few races even if you can't afford to upgrade anything after buying it. Then keep upgrading it throughout Bogmire. Once you get used to the track layouts, you'll have no problem.
Nothing changes here, just new tracks and better enemies. Start saving up for the Havac, but don't be afraid to pump up a little more now if you think you'll need it to survive. You probably don't need to upgrade any of your equipment to level 4 to make it, but level 3 is OK.
The Havac is to Nho as the Battle Trak is to Bogmire, but it's not as important to upgrade now, since your Battle Trak is probably pumped up and it still races well on the ice. It's still not a bad idea to upgrade even with little money for upgrades, though, since the Havac won't need a super engine to get through the snow, and it'll go even faster with a super engine. :) If you're playing Snake, don't upgrade your engine too much right off. You're used to the Battle Trak now; the Havac will go too fast and you will run into corners... your lack of cornering skill becomes more problematic at high speeds. You can beat the opposition easily enough, so don't rush to upgrade the engine too soon.
This is it! The final planet. You can't even access it in Vs. Mode, so if you've been toying around with that, this will be a whole new adventure for you...
This is where the game gets tough. You may well find it somewhat comparable to your struggle on Chem IV way back at the beginning of the game, but it won't quite be that bad. On some tracks there are reverse warp arrows which send you rocketing backward, probably costing you more victories than anything else. Your opponents will all be sporting Havacs, and they'll be powerful, so you will no longer have the technological edge. That means relatively minor mistakes become much more serious here, and this why hitting a reverse warp arrow can ruin the race for you. On top of all this, the track layouts can be somewhat confusing.
If you get more than the required points in Division A before the end there's no reason to continue racing. Go directly to Captain Braddock and watch the ending.