Where to Stay:
For a more budget friendly place look into La Casa de Carmen, which is near the corner of Via España and Via Brazil. They have a really nice homey/hostel fee. They aren't well located, but it is a $1.50 taxi to the Calle Uraguy are where you can find lots of restaurants and bars. And the price is $14 for the dormitory and $30-50 for a private room. Many Europeans can been found staying here.
If you want to stay in the old, more culturally interesting part of The City look into Luna's Castle. This hostel has dormitory space only for $12 a person per night. A young set of mostly Americans is who your likely to share your room with. If you want a quiet place to stay, go to Casa del Carmen. This is more of a party place. They are working on installing private rooms on the third floor. Their ETA is January 2008.
Instead head over to the El Mercado de Mariscos (Seafood Market) at the end of Balboa. Upstairs is a restaurant that is decently priced ($6-9 a plate) with really good seafood. Go for lunch or early dinner so you can see the fresh catches they are selling. You must try the civiche, a Peruvian dish made of marinated raw seafood. Jeff recommends the Fillete de Macho. I guess this restaurant isn't great for vegetarians either.
Another good restaurant is Pizzeria Athens for Greek lobsodosomo, and Imam pizza. The food is excellent and the beer is not overpriced. This is a good restaurant for vegetarians. They are on Calle Uriguay and Calle Cincuenta. There is a gas station right in front of them. If the taxi driver doesn't know where it is, tell him it across from Elite and Hooters. They always seem to know where the strip clubs are. I believe they are closed Wednesday nights, Athens not the strip clubs. I don't know about the strip clubs.
For fancy, but not overly priced Italian try Rino's near Calle Uriguay. The building looks like a huge white mansion. Wear long sleeves though because it is always chilly in there. They have a delicious maracuyá mouse($2.50) and lasagna ($6.00).
If you want to get some travel recuerdos (souvenirs), try the YMCA Mercado in the neighborhood of Balboa. There are two big buildings with all different types of indigenous and Criollo crafts at good prices. It is a group of individual vendor booths, so often you are buying directly from the artisan.