Places with lots to see by walking around:
Balboa Park: the Zoo, gardens, museums. Not really any hotels in that area that I know of, unless you stay downtown ($$$).
Old Town / Presidio: sort of a preserved Old West part of town, but also with touristy bars and restaurants. Right next to the airport. Also has a trolley stop. There are a few hotels around there that are not too crazy expensive, for example this Best Western:
http://www.haciendahotel-oldtown.com/
Downtown Waterfront / Embarcadero / Convention Center / Ballpark: Old ships and maritime museums, the Midway battleship, also with touristy bars and restaurants. But not far from Little Italy / India Street / Columbia Street, with good bars and restaurants. Lots of trolley stops. Hotels in that area are generally $$$$
Coronado Island: IMO best beaches in San Diego, some decent (if pricey) restaurants; can take the ferry from Embarcadero across the bay to here. There are hotels in Coronado but I am sure they are $$$
La Jolla: City just to the north of San Diego. Nice beaches, some decent restaurants, great ocean views. The kind of place you walk around and say to yourself, "once I win the Lotto." Most hotels are $$$$ but I may know a few secret cheap ones.
Then there are some less touristy, more "known to the locals" type areas in the city...
Ocean Beach: the land that time forgot, an old-school 60's / 70's California beach town right in the city. Has a nice pier, a decent beach, usually good surfing, and Sunset Cliffs are cool for hiking, tidepooling, etc. Hodad's and a Pizza Port are here. Don't know what the hotels here are like, mostly they are hostels.
North Park: was a dump 5 years ago, now undergoing a kinda hipster-revival, good pubs. Just to the North side of Balboa Park. Probably some hotel deals around there (Motel 6 kinda stuff)
Hillcrest: just west of North Park, just North of downtown; best restaurants in the city IMO.
Then there are areas that all the tourists know about, but to me are worthless, like:
Mission Bay: an artificial bay with poor tidal flow, nasty stinky water, fake beaches, expensive hotels, drunk college kids from Pacific Beach and Sea World. No thanks.
Gaslamp District: Downtown; bars, nightclubs, live music and restaurants...but IMO too touristy and overpriced--not nearly as cool as it used to be.
BTW, the Town and Country is on Hotel Circle, which is a big loop of hotels where the I-8 and I-5 intersect. It is kinda close to the Presidio and to Hillcrest, but other than malls there's really nothing to see at Hotel Circle. BUT, it is pretty well centrally located to all these other areas and attractions, and you can pick up the trolley on Hotel Circle. And there's a range of price and quality from the T&C (which is near the top), to Holiday Inn, to Mom and Pop places, to the Motel 6-type places (and worse). So Hotel Circle is kind of the "default" place to look for lodging.
Hope that helps. Let me know what sights you're thinking of seeing and maybe I can give more specific help on your hotel choices.
(EDIT: I see kev211 ninja'd me; I concur with him)