Army housing on O'ahu is split into north and south regions. Being assigned to Tripler, you'll be in the south region, which means AMR, Ft. Shafter, and TAMC quarters could be offered to you. Unless you have children, AMR is probably not an option.
A few years ago, Army housing in Hawai'i was a joke, and that is still largely the case (I lived in Army quarters for the first two decades of my life, so I have some perspective here). This fact is even further driven home because the Army housing exists in such stark contrast to the relatively nice Navy, Air Force, and Marine housing across the island. They're in the process of tearing down and rebuilding much of the existing Army quarters. However, the process has gone painfully slowly and it is principally focused on Schofield Barracks (north housing region). The only new quarters in the south region are currently on Ft. Shafter, and they are sufficiently scarce that you can't realistically expect to be offered those.
The overwhelming majority of TAMC physicians live in off-post housing, but there is no favorite spot. I'll try to give a brief description of the most popular choices.
1) West O'ahu - this is a broad area with several choices. The chief advantage of this area is that the real estate is relatively cheap. You can probably afford a single-family home in this area. The climate is appreciably warmer and drier, so air conditioning (preferably central) is a must, in my mind. The major downside is the traffic, which is not to be underestimated.
A) Waipahu/Kunia - slightly closer to TAMC and H1, but also slightly more expensive. Highly commercialized.
B) Ewa Beach - cheaper and more residential, but there's the added hassle of dealing with traffice on Ft. Weaver Road (can double your commute time from 1 hour to two).
C) Kapolei/Makakilo - a little farther away and a little cheaper. Ko'olina is here, which is a resort with many townhomes.
2) Mililani - a very nice town with the added convenience of Schofield Barracks being nearby (commissary, cheap gas, etc.). A higher elevation = a cooler climate. Greater access to the North Shore, but it's probably the farthest away from the beach you can live on the island. Traffic is also an issue, nearly as much of one as living in West O'ahu.
3) Aiea/Pearl City/Salt Lake - much closer to TAMC with an expected increase in cost and congestion. I know very few people that live in this area, and for good reason - the benefit in commute does not seem to outweigh the added congestion. You might as well live in Downtown or Waikiki where there are actually things to do.
4) Downtown/Waikiki - mostly preferred by single physicians who want to be close to the restaurants and nightlife that this area offers. This is obligatory condo/apartment living.
5) Hawaii Kai - on the whole, probably the nicest town on the island. Condo/apartment prices are high, but not out of your price range necessarily. The major downside is that you have to go all the way through the full length of Honolulu for the commute. I've known four people to live here, two of which moved because of the commute.
6) Windward O'ahu - less commercialized side of the island. Feels very far from Honolulu thanks to the intervening mountains, despite only being about 10 miles away, which can be both a good and bad thing. Commute to TAMC is very doable thanks to H3. The MCBH offers most of the benefits of a nearby military base without having to go over the Ko'olau mountains. More expensive, but not prohibitively so.
A) Kailua - relatively expensive. Doubt you could afford a single-family home there without supplemental income (spouse, family, etc.). No hotels (by city ordnance) but still relatively crowded by day. Kailua and Lanikai beaches are among the best in the world.
B) Kaneohe - slightly larger (more restaurants; has a mall, movie theater, etc.), more crowded, and cheaper than Kailua, but not as quaint. You might be able to afford a home there.
7) Waipio/Waimanalo/Waianae - I do not recommend living in these towns.
I hope that helps.
