For the first time I see problems with the DS. Thankfully, it's in a good way.
The first mode anyone playing this should do is the single-player. If you are new to the Mario Kart series, especially if you are new to the portable, do the Missions first. These are small exercises designed to improve your racing abilities. They also allow you to experiment with a variety of drivers.
From there, I started immediately on the Grand Prix, where I think the heart of the game is. There are 3 engine classes, 50cc, 100cc, and 150cc. 50cc is the easiest, 150cc the hardest. Each engine class has a Nitro Prix and a Retro Prix. Each Prix has 4 cups, and each cup has 4 races. That's alot of tracks, and they tend to get a little bit harder as you go through them. The Nitro tracks are all brand new, only on the DS game. The Retro tracks could be from the SNES, N64, GBA, or GCN versions of Mario Kart. Each cup has one from each game.
After finishing both Prix on 50cc, I decided to set-up the WiFi connection. This couldn't have gone much easier. You simply pick WFC on the Main menu, setup a connection, and start playing. You have 4 different ways to choose who you want to race. There's the Worldwide option, the Regional option, the Friends option, and the Rivals option. Worldwide and Regional both work right away, but you need to have raced a few times to get friends or rivals. Once you get some friend codes from people you know(you can't get these through the game), you can chose the Friends option. I'm still not sure how the Rivals option works.
Pick one of the options to pick your opponents. This may take a minute or two, depending on when you are playing and how many other people are online. However, on nintendowifi.com, they have stats showing when the online traffic is the most, and therefore when the best times to play would be. After getting a group together, you simply pick your character and kart like normal. When you get to the track selection, pick one track. Each other person will pick one, and the computer selects one at random. If there is more than one person who picks a track, that track is automatically chosen. You then race 4 times, with different tracks each time. At the end, a winner is chosen. You can then choose to leave and find another group, or stay for another round. Stats are recorded on individual races. So, say if you win 2 races, but screw up on the other two and come in 3rd, your record is 2-2, not 0-1 for losing the entire race.
Overall, this game is amazing. Even having played it for only 2 days, I am already addicted, and can't wait until I can unlock even more new characters, items, and who knows what else. This game may seem simple on the outside, but with so many different possible combinations, nothing is ever for sure.