Hi,
I grew up always within a 20 minute drive to the base. Tacoma, Spanaway, Parkland, ect... all have a bit of a bad repuatation. Tacoma really was so bad twenty years ago that there was a special federal task force to take care of the drug, gang, and crime rate. However, things are a lot better now.
Here is where I would avoid: Tilicum and most of Lakewood. There is a lot of attractive sounding housing right by the base. When my hubby and I first married, we got a very cheap, older, but pleasant apartment there. My brother in law came to visit and said "HEY! I know this place! I've seen it on COPS!". Sure enough, in the three-months we lived there, they Cops TV show people were there at 3 a.m. one morning (thanks to our lovely neighbors 5 doors down).
Fife is okay, but I wouldn't choose to live there. It is further away, not as community minded, and more expensive.
The East side of Tacoma and Parkland/parts of Spanaway were the areas targeted by the task force. They still are rampant in poverty and would be the less desirable places to end up. (Also the cheaper ones naturally.) As you progress down Pacific Avenue/HWY 7 away from Parkland and further into Spanaway, it gets nicer. Most of the areas in the Bethel School District are nice decent areas (and the school district is fairly good).
Puyallup is a very nice place, but further from the base and much more expensive (they also have the best schools).
North Tacoma is a nicer area. It's a mix of good and bad, but traditionally has been a cleaner, safer area. I would live there, but only if it is convenient to where you are going every day. Traffic in the Pierce county area is something to be avoided!! There is a lot of value to live near where you work or go to school in the area.
My advice, if you can find a good rental would be a small "town" called Frederickson. It is the neighborhood I spent most of my childhood in (we moved away from Eastside Tacoma when I was 10). At the time, it was just an un-incorporated area of the county, it is way over-built up. The residents formed a city in order to stop some of the over building. However, a lot of small businesses have moved in and it is very close to the base once you know the back roads. The crime rate is low, and it is a pleasant area to live in. It is in the Bethel School District.
If you live in Frederickson, you can drive down 176 to Spanaway and then down Pacific Avenue to Military Road (by Sprinker Rec Center). If you go down Military Road, it is actually a civilian use road that drives between Ft. Lewis and McChord AFB. Using this you can be on the base in less than 1/2 hour from Frederickson.
Best of luck!! And, welcome to Washington. It does rain a lot, but it makes the area so very beautiful! And make sure you check out Pike's Place Market in Seattle and plan on two days to the Puyallup Fair in September. LOL