Anywhere near the Power & Light district is pretty good as they have been renovating downtown heavily. The Plaza, Westport, Brookside and Waldo are all pretty nice neighborhoods within Kansas City proper, and they are all pretty close to the Power and Light district; yet all are fairly expensive. Apartments on Wornall, Holmes and State Line are all pretty nice as well.
Avoid streets like the Paseo and Troost, or anything east of Troost Avenue. Troost Ave. has been the historic dividing line of proper, affluent and white Kansas City whereas the majority of the ghettos in Kansas City are found east of Troost. Its sad, but true.
As far as the music scene goes, welcome to hell if you are into progressive and power metal. There are a few good venues in Kansas City but the only one I have ever been to is the Uptown Theatre (DT, Queensryche and Fates Warning/G3/Ted Nugent).
Luckily for you, Lawrence, KS is only about 25-30 minutes away. Personal experience has found that I have seen more metal shows here than I ever did in Kansas City (lived in the KC 'burbs since 1994). Behemoth, Gojira, Between the Buried and Me, Behold the Arctopus have been to Lawrence inside of one college semester, and I hear that Scale the Summit will be here in March.
Hope you enjoy your time in KC. Its mainly about finding a neighborhood that suits you and being able to find stuff that fits your personality and interests. If you are the outdoorsy-type, you'll love the spring and summer with the abundant hiking/fishing/camping areas anywhere from in the 'burbs to a few hours outside the metro area.
