The career mode begins with an extended series of FMV sequences that set up the story, such as it is. During these cutscenes, you have to win three races, which is not hard to do. The first two races are Sprints and then a 2-lap Circuit. Your "hot" BMW will barely break 140 and the other cars are much slower--consider this your tutorial to get you used to racing. You shouldn't have any trouble beating the races.
After those three races are won, you start a race against Razor. Don't worry too much about it, you won't be finishing this race. More FMV sequences set up the beginning of the main story: you find yourself with 30,000 cash in your pocket (40,000 if you have an NFS Underground 2 career save on your system) and a small selection of cars to get you started on the racing circuit again.
If you have the extra money from playing NFSU2, buy a Golf GTi, it's the best starter car. Otherwise you want the Lexus IS300 or the Fiat Punto if you want good handling. The Chevy Cobalt SS is faster, but handling is poor. Once you've selected your car, you have to start working your way up the Blacklist.
NOTE: "CP" in the race descriptions means "cop probability". Maximum means the cops will almost assuredly be chasing you during the race. Minimum means you can breathe easy--no cops to worry about.