Prices & demand show the North End's Proctor district as being one of the most desirable locations, but that doesn't mean it's the only "good" place to live... There are some great locations in Central and South Tacoma that can give you good accessibility to shopping, restaurants, the freeway, etc. And if you wanted a historic home for a reasonable price I would certainly broaden your search to include these locations. Your architectural preferences will inevitably guide you... if you want to live in a rambler or new construction, there are particular areas that are going to offer you those options.
I agree with the others about doing your own research (or ask an agent to give you an overview if you're not comfortable with the area). But don't rule out neighborhoods just on word of mouth - see for yourself. Do visits at different times of the day and on the weekends. Take a walk and talk to the neighbors. Go on the City of Tacoma's crime website and do the hard research http://(www.tacomacrime.org). The variety and diversity is what makes Tacoma a great place to live -- there is something for everyone!
6th Avenue is starting to boom now, too, east of Union. It's technically on the edge of "Hilltop" area, but it's vibrant if you're young. An apartment or townhouse in the area should be affordable. And on the North side of the street to the water you start to get into the "Old Town" area, so there's plenty of room for variety in the area.
More traditional areas of the town continue to be popular, such as the North End, from Proctor into "Old Town." These neighborhoods would include larger, older houses, and corner hangouts at the intersections of major arterials. There are some very nice neighborhoods in the West side of town, along the water, around the Narrows, all the way to say, 12th and Stevens, which is about where you start to transition into the Central District.
The Central and South Districts have some pretty bad neighborhoods, but there are also some perfectly nice little enclaves tucked into the middle of these areas that might be a good bargain for the less timid.
I am a native of the area and can give you lots of info! Tacoma is a very accessable place, and you can get from the north end to the south in a few short minutes. Give me a call and we can talk specifics!
It really depends on what is important to you. Aaron may have a good resource for you and you can also work with an agent who both works and lives in some of the area. I've never lived in Tacoma, so I wouldn't be your best resource but if you were considering neighborhoods on the eastside of Seattle, I could tell you lots.
If you need a referral to a native Tacoma agent, let me know.