Shenmue 1 and 2 are awesome, and there are a few other games that are great like Daytona USA 2000, but overall the Dreamcast is very underwhelming and by far Sega's worst console. The controller is one of the worst ever conceived. Fortunately if you buy the Total Control 3 you can use your Saturn pads.
You are out of your f*cking mind. The Dreamcast is amazing. The controler...I can't hate on it enough though. It sucks terribly. The worst Sega controller ever? No question. Worst system, not even close. Probably the best if not for the Saturn, and even that is a tough call.
Great DC games (too many to list, seriously).
A slew of fighter ports, usually with zero load time. Jojo, Alpha 3, All three versions of Street Fighter III, Marvel vs Capcom 2, and Capcom versus SNK. Skip CvS1, and CvS Pro. They suck miserably. Also, Tech Romancer, Powerstone 1, and 2, Project Justice. Damn, such a great fighter system. Then there are some SNK ports as well, although they didn't turn out as well. KOF 98-2002 were nice. If you love Sega AM-style cheese, don't skip Fighting Vipers 2. If you like Firepro, Firepro D is still one of the best versions, and its *insanely* user hackable.
Shooters galore: Twinke Star Sprites (an awesome sort of versus shooter/puzzle thing. Very unique), Ikaruga, Mars Matrix, Boarder Down, Under Defeat (wonderful!), and my personal favorite Rez. There are about three dozen more, but these are the good ones.
Sega GT is still better than any version of Gran Turismo, and the third best Turismo clone out there. Other racing games were f*cking great as well. Test Drive LeMans (with true 24 hour mode, and 1/16 the load time of the PS2 version), Test Drive V-Rally, Runabout, Shoutoku Battle (I forget what US name they used for this), and then of course there is Crazy Taxi, which is really awesome despite it being something we are all sick of, and it costing $0.49 most places now. Ferrari F355 is the most hardcore racing game ever made...except maybe Grand Prix Legends. Daytona 2 is RAD!!! but requires, seriously, *requires* that you have a racing wheel or the car tends to smack a wall the second you even touch the control stick. This is actually the programmers fault, and not the sucky controller fault because other racing games play great with the DC pad, they will just cramp you hands big-time.
Other stuff: Sakura Wars 1-4. 1, and 2 are just ports of the SS games, but SW 3 is one of my favorite games ever. A truly beautiful and huge creation. Shenmue 1 and 2 (1 is rough, but 2 fixed virtually everything and blows my mind). Any true Sega fan knows the total uniqueness of SeGaGaGa. I get misty-eyed just thinking of it.
Oh yeah, VIRTUAL ON: ORATORIO TANGRAM. My Dog this is the best f*cking thing ever. It basically cannot be played on an actual DC pad though, so get a Total Control and use a Saturn Twinstick. There are Twinstick controllers for DC, but they are more expensive, less durable, and not as cool looking. Until you have linked two DCs each fitted with VGA displays, Twinsticks, and VOOT, you just haven't played a video game. I'm sorry, you lose.
Then you have your rather weird shit, one of my favorite genres: Roomania, Cool Cool Toon, Seaman, and Lack of Love (which I confess to having never played, but it looks cool, and no its not a chick sim)
Check it out:
Odds and ends: Sports Jam, Samba de Amigo (requires expensive maraca controllers), Gundam 0079, Stupid Invaders, and Skies of Arcadia (a very great RPG). If you have good Japanese skills, Ghirens Greed is one of the greatest strategy games ever. Its based on Mobile Suit Gundam. Its basically the same as the Saturn version but with near-DVD-quality videos this time. Check out the NES, 2600, and SMS emulators. They are *really* good.
Things to avoid: American shit. Any US published wrestling title, although there are several OK JP ones, and one great one (Firepro), Anything with Sonic in it. Sorry, but I litterally get sick trying to play any Sonic game in 3D. Confusing as shit, and twice as frustrating. Staring at a wall+desperatly trying to fix a permanently broken camera=you missed the point of Sonic. Avoid the chick sims, also Chaos Field. That thing sucks.
You NEED: a controller adapter so you can use Saturn pads, or shit, even PS1 pads. Yeah, the DC is so bad that PS pads are a dream by comparison. Sadly stuff like Shenmue, and Rez require an actual DC controller, but most of the fighters and shooters will work with an arcade stick, which means they will work with a SS pad if an adapter is used. If you want to get into the racers, and wheel isn't %100 required unless you want to play Daytona 2, but its nice for F355, and Le Mans.
You WANT: A VGA box and shitloads of memory cards at least one of which is a real Sega VMU/VMS with the LCD.
I know I missed a bunch of stuff.