Good thing I came out of hibernation to check in today, and saw the news that the Captain is belatedly heading this way!
The Home Course and the occasional round at Gold Mountain-Olympic (but don't get stuck in the Narrows Bridge traffic) are far and away your best bets. You will learn quickly when and when not to venture onto I-5 to get around, as the traffic into and off JBLM at certain times of the day and week is unbearable. Stick to the courses on base on those days.
Expanding the perimeter a bit, I have a fondness for the Classic Golf Club as well, the Moore-family owned property in Spanaway. Very solid track and very traditional PNW-type tree-lined track but with some intriguing green complexes.
Also, Washington National and Druids Glen, depending on course conditions, can also be pleasant diversions, but would not be worth your regular plays, IMO...Lake Spanaway golf course is also quite pleasant.
Congrats!