For street racing you probably will want Gran Turismo 3 on the PS2. I
have an Xbox, and so far the only good racing games for it are NASCAR
Heat, F1 2001, Rallisport Challenge, and Project Gotham Racing. Of
those four the only street racing game is PGR, but even though it has a
pretty realistic feel to it many of the tracks are very narrow and
tricky, not really conducive to good head-to-head racing. Rallisport
Challenge OTOH is a total blast, even if you don't think you are into
rally racing you will probably like it. But there are rally games for
the PS2 also. For the Xbox there is also Test Drive and a couple other
similarly forgettable ones which should only be bought by people who
like to crash cars, and that's about it for now. Eventually Pro Race
Driver and some version of Need for Speed will be available for Xbox.
Also said to be on the way are Sega GT 2002 (similar to GT3 but probably
not as realistic), and factory sims for Mercedes Benz, Lotus, and
Lamborghini. Impossible to predict what the quality of those titles
will be.