Alan,
a lot of people her (including myself) like Fuji Neopan 400, but you should get good results from any conventional 400 ASA B&W film like Kodak TriX (the traditional choice), Ilford HP5+ and Agfa APX400; if you shoot a lot in bright light, you might also consider APX100, or Ilford FP4+, or one of the budget films from Eastern European makers, like Fomapan 100 or Efke KB100 (eg. from JandCPhotography). For low light, I like Fuji Neopan 1600, and for 'available darkness' (indoor shooting with artificial light), Kodak TMax 3200.
If you are still new to film developing, you might want to stay away from the flat-crystal/T-grain types of B&W films (Kodak TMax and Ilford Delta), because they are much more finicky when it comes to exposing (esp. with street shooting, where you don't always have time for a perfect light-reading) and development; these are more fine-grained, so you don't get the traditional gritty street look, either.
Until you are starting to develop yourself, you might want to try the so-called C41 B&W films (which are, essentially, color negative films that produce monochrome pics), like Ilford XP2, Kodak 400BW (or whatever it is called these days), and Konica VX400 Monochrome - these can be developed by any color lab; they don't give as high apparent sharpness (esp. edge sharpness and accutance) as traditional B&W films), but on the other hand, skin-tones will be very smooth, results will be fine-grained, and the great plus: you can expose them from around 100 to 800 ASA on the same roll without changing development (can't be doen w/ traditional films!)
Roman