Dragon Ball Z: Budokai Tenkaichi 3 is a fighting game, that much is obvious. But there are several different modes that you can play it in. Dragon History is where you play to unlock more characters, earn Z points, and most importantly, unlock new skills. You'll start with the Saiyan Saga. After you beat it, you'll unlock more sagas to play. Each saga consists of at least three battles. The battles can have many parts, so watch the screen carefully. When it is time to switch to a new character or trigger an event, you'll see (1) on the bottom of the screen in the middle. Press (1) to continue to the next part.
Ultimate Battle is another mode of play available. It is a "survival game" challenge. Inside it, you'll train and battle to improve your skills and earn Z points. The biggest challenge is in winning all the battles.
The Dragon World Tour is where you go to play in tournaments. The World Tour is "for one player, and will open at a specific time," to quote the game. But if you don't feel like waiting on a specific tournament to be open, you can choose Free and play alone or against your friends. Just be careful because in tournaments, you can be eliminated instantly with a "ring out."
If you want to just battle against your friends or practice against different characters, you can go into Duel. There are three different Duels that you can choose from. Single Battle allows you to each pick a character and fight. It's a quick and easy way to match up against your friends. In Team Battle, you'll pick five fighters to battle with. Obviously since it is 5 on 5, Team Battle is going to last a lot longer. The last battle type you can choose is DP Battle. DP stands for Destruction Points. Basically you form a team compromised of characters that total no more than 10, 15, or 20 (whichever you choose) DPs. Each character is assigned DPs based on how strong they are. It will tell you how many DPs a character has on the character selection screen. Just in case you're curious, of the characters you can choose from at the beginning (because I'm not going to revel who you can unlock), Hercule is the lowest at 1 and Super 17 and Syn Shenron are the strongest at 9. If you want to use up your DPs on less than 5 characters, you can.
If you want to go online and compete against someone outside your TV room, you can do so in Dragon Net Battle. All you need to do is set up your wi-fi connection. It will search for random people for you to compete against or you can choose to battle anyone that you have set up as a friend. You set your friends up under Manage Friends.
To spend your Z points and buy things to customize your characters, you go into the Evolution Z selection. Launch is the shop keeper, so be forewarned that you never know if she's going to be nice or mean when you go in. There's several more game choices, so as you can see, there's a lot of different ways to play.