I have been living in Santa Barbara for a while now, here is a list of things to visit/do:

1. go look at the Old Mission Santa Barbara,
2. check out the inside of the Arlington Theatre (you can go inside when no movie is playing),
3. take a walk in the Santa Barbara Public Market and try the almond brioche at Good Bread
4. try the macaroons at Renaud's Patisserie & Bistro in front of the Arlington Theater
5. try the gelato at Spoon
6. visit the Santa Barbara Courthouse Docent Tours and climb on top of the tower
7. check out State St and all the cute/colorful courts and shops
8. take a look at Casa de la Guerra
9. try the coffee at Handlebar Coffee Roasters
10. walk down to Stearns Wharf pier
11. have lunch with your feet in the sand at Shoreline Beach Cafe
12. walk around the boats at the Santa Barbara Harbor
    
13. go to Shoreline Park and watch the sunset from the beach
14. go play some beachvolley at East Beach
15. walk on the sand to Butterfly Beach
16. go get a drink at sunset at Belmond El Encanto

There is a lot more to do (paddle boarding, hiking, zoo, etc), but if you only have one day, this should be plenty. :)

Here's the few I make sure to always do when I'm up there:

Brunch at the Four Seasons Resort The Biltmore Santa Barbara.  Dinner at San Ysidro Ranch.  Cocktail at Joe's Cafe.  Any meal at Hendry's Beach.  Singing, card tricks, and the cajun martini at The Palace Grill.

If you want to be a tourist and see a lot fast, hop on the Landshark with  Land & Sea Tours. They take you around town on an open air amphibious vehicle and then into the harbor -- boat ride! It launches a few times a day from Cabrillo Street (not to be confused with Carrillo or Castillo streets) by Stearns Wharf. 

Bike along the waterfront. Maybe bike into Montecito and have a meal at the Four Seasons Resort The Biltmore Santa Barbara 

As Andreas said, wine tasting is great.

I also recommend a stop at the Santa Barbara Public Market. If nothing else, have either Enjoy Cupcakes, Santa Barbara or Rori's ice cream while there. Or consider picking up a picnic's worth of stuff -- fresh break, cheese, salami, and again, cupcakes, and take your treats to the beach or a pretty spot like Alice Keck Park Memorial Gardens

If it's a pretty day (and when is it not?), go to the top of the Santa Barbara Courthouse Docent Tours to take in the view of the red tiled roofs and snap some photos to make your friends jealous.

I would go wine tasting in The Funk Zone (off of state Street by the train tracks) . Lots of nice wine bars with tasting menus.

With a bike you can travel almost everywhere in Santa Barbara. Start with a cruise down State Street Route to the beach, turn left, and ride along the beach bike path to the beach volleyball courts. The path continues on the other side of the street, winding around the bird sanctuary and zoo, then crosses past the cemetery, and continues on the ocean-side of Ty Warner's mansion, all the way to Butterfly Beach and the Four Seasons Resort The Biltmore Santa Barbara and Coral Casino Beach & Cabana Club. You can do the round trip in about one hour.