Planning on a week or a little more with a side trip or two.
Prefer to stay near the equivalent of old town.
Prefer an area not filled with bars and night clubs. Not the heart of night tourist life.
Want to be near downtown or very easy Metro access.
Restaurants nearby important especially for breakfast.
Parks not unwelcome!
We like walking but not big hills!

So what areas do you recommend? Is the west side Buda and the right Pest?
Jewish quarter?
Area near government buildings?
Museums and other cultural buildings, museums, symphony, etc. OK.

In Prague we stayed in a quiet pretty area with a park and two Metro stops from downtown and it was fine.

We won't want a big hotel but will seek a rental.

Any suggestions welcome re districts and more.

We stayed here http://www.bellevuebudapest.com/ years ago, and an online search indicates that it still gets good reviews. When we stayed, breakfast was very light.

BudaBaB - BudaBaB

I prefer to stay on the Pest side as there are more options for eating and sightseeing. I have stayed in a couple of hotels but highly recommend the apartments from this outfit:

Six Very Special Apartments in the Heart of Budapest

The full service Marriott in Pest is wonderful with views of the river and the Buda side with its Chain Bridge, Castle, etc.

You used to be able to get it on Priceline.com for about $70 usd per night. Don't know what it goes for now, but great location! And a great hotel, recently renovated.

We also stayed on the Pest side-it was flatter and generally more density of sights, shops and restaurants. We stayed at the 7 Seasons Apartments Budapest. They have their own site: 7Seasons Apartments Budapest | Apartment Accommodation in Budapest but we used booking.com. We were driving, so chose them partly because they had underground parking and could just park the car and not touch it again until we were ready to leave town.

Who's we?

We stayed in the K+K Opera ages ago and it had a great breakfast spread and nice modern little rooms. Plus it was on a side street thisclose to the Opera House and 40m from Andrassy so it was smack in the middle of wherever you wanted to go in Pest (which has most of the main sites other than Gellert Hill and the Castle district). And of course, the metro stopped right near at the Opera station.

"Is the west side Buda and the right Pest?"

The west side of the Danube is Buda, the east is Pest. Budapest is a portmanteau of the names of the two towns that developed on opposite sides of the river.