Hi Welcome to the forum. I would personally suggest that you look into the downtown area on the east side. Los Yoses is a good area, very close to the University of Costa Rica. Barrio Dent is also similar. These areas were built up 25 years ago and each house has a unique architecture.
Another possibility would be near the La Sabana Park, say in Rohrmoser, and the Sabanas (Norte,Oeste, Sur). There are new condominiums going into that area, so activity will pick up in the near future I imagine. In any case there are already lots of restaurants, museums and activities in the immediate area.
Either area would put you right in the middle of town, where you can have easy access to everything.
Have a look at the American-European web site for additional information about
Places to Live in Costa Rica