I would recommend staying in the downtown Seattle area. There are many fine hotels there but the prices can be a little cost prohibitive. I believe it's the Edgewater that is actually built out over pudget sound. The pic of the Beatles fishing out of their hotel room was taken there. There is also a Marriott along the shoreline there that is very nice.
We went cheap after an Alaskan cruise as stayed at the Springhill Suites, which is a little north of Downtown. It was a nice room and they offered a shuttle service that would take you to Westlake Mall, the convention center and the Space Needle. I was NOT overly impressed with the shuttle service at Springhilll Suites when when we were there.
There are several very nice restaurants on Lake Uniion, many with views. I believe the name of the restaurant we ate at there was Chesters Crab House. Very good food, good service, nice atmosphere and I felt the prices were reasonable. Most of the rest of the restaurants we ate at were chain restaurants with the exception of one "tea room" which was in part of the old "underground." It was an interesting place and served good sandwiches and coffee (Seattle seems to be all about the coffee). Unfortunately I can't recall the name. If you do the underground tour I'm certain one of the tour guides will point you that direction. That's how we found it ourselves.
In the downtown area you won't really need a car. Most things are walkable (IMO) and if you're feet get tired, most of the downtown area is in the bus systems "free zone." I found the bus drivers very helpful in telling me which bus to take and where to get off. Of course, I've always found people willing to tell me where to get off at.
We waited until we got to town and then I rented a car. There are several locations downtown and the prices were reasonable. I only rented a car for the time days I needed it and we dropped it off at the airport location (no additional charge) rather than take a taxi, town car or shuttle service to the airport. It was much more convenient for us to do it that way.
Mt. Rainer is a wonderful day trip but I really wish we had spent at least one night out there and spent two full days exploring all there was to see. Next time I plan on doing that. We found the undergound tour interesting and worth the money. They told a lot about the history of Seattle. Don't buy the books they recomend (Son's of the profits was one of them). I did not find them particularly well written or a very good read. We did not go to the Space Needle. Wife felt that was just a little to touristy and we've done the Strat in Vegas. The restaurant at the top of the needle doesn't generally get very high marks for food or service but it's the experience and the view people agree makes it worth it. Personally I'd rather have better quality food and service. I can get the view without the restaurant.