I've lived downtown and the old southwest for about 15 years. I love it downtown but it definitely is probably not for everybody. Old SW and the Newlands are great neighborhoods where you can get the best of both worlds, walking distance to downtown, midtown and the river but great, established old neighborhoods with lots of character.
Both those neighborhoods however can be tough to get into as people tend not to leave, but you should be able to find something in your budget.
Otherwise I'd check out Galena, Windy Hill or if you want be closer in somewhere in the west Moana/Plumb-Skyline area. One good thing about Reno is everything for the most part is pretty close, and while it has bad areas, you'll generally know it when you see it.
Welcome to the biggest little city, it's quirky and unique, people and lifestyle are great, you're going to love it here.