What's the best neighborhood in Detroit for a first time visitor?

You definitely want to stay downtown or near downtown in areas like Corktown or New Center. These are the hubs of Detroit and you will experience Detroit, both new and old if you stay near these areas. If you have to stay outside Detroit, stay in Royal Oak (Michigan) which is about a 10min (or less) drive away from downtown Detroit. Depending on who you ask, "locals" tend to stay at one of the casino hotels like Greektown Casino-Hotel or at Detroit Marriott at the Renaissance Center (The Ren Cen). The Atheneum is also popular. These can be expensive though (by "oldskool" Detroit standards). Trendy hostels have popped up recently too, so you might want to check into those. Just make sure you are Downtown, Corktown, or New Center area.  

You could actually even stay in Windsor (Ontario)--just across the river from Detroit, and exchange rate would give you a bargain room rate! Just make sure you are in one of the Riverfront hotels (Caesars Windsor is quite deluxe, but affordable). Some cool nightlife/ clubs--Deadmaus is playing Windsor in the fall). Windsor is also 100% safe, and there is a tunnel bus that takes you right to downtown Detroit.

I would recommend staying in Downtown or Greektown. Greektown is fantastic as it has a nice atmosphere throughout the day and into the evening. The Greektown Casino-Hotel is nice and is centrally located to the casino and all the delicious restaurants in the area. Just a short 10-15 minute walk to central Downtown (Woodward Ave) and close to theaters up Woodward as well.