We stayed at Welton Centrum Hotel. A nice hotel placed centrally in the old city. The rooms were more than good enough, and the breakfast was excellent. The hotel had it's own spa with a sauna, Jacuzzi and pool. We didn't try any of the treatments, but it was really good to just soak up in the pool after a long day walking.
Riga isn't a really big city, and almost everything is within a 15 minutes walking distance (that be if you walk the right way, and don't walk in circles for almost an hour....). The old city is filled with nice architecture, and you should really take some time to walk around here. There is also an museum (free of charge) who is worth a visit if you are interested in Northeast-European history.