Hey Kris
I grew up all over that area. Rent the car - LA is NOT like New York where there's a cab every 10 seconds. And, yeah, catching a cab for 45 miles (and 2 hours with traffic) will add up REAL quick!

The Groves is nice, as put before - it's overshadowing the Farmer's market, but both are within about a block of each other. Eat at the Market, shop at the Groves. You could also keep venturing along the 10west to Santa Monica, where the Third Street Promenade is. Or, catch 405 North 2 miles and get off at Westwood, and catch the town and UCLA's campus if you feel inclined. All three of these are within, probably, 15 miles of each other.

Good suggestions about downtown. Don't forget Alvera St., the original neighborhood where LA started, along with cheesy Mexican gifts and good Mexican food. It's about 6 blocks from Chinatown. You can catch a Dodger's baseball game while there, since I think the season will be in full swing by then. Also, you can catch the Red Line (a local commuting train line) at Union Station (about a mile from the convention center) and take it out to Pasadena, where Colorado Blvd is nice to shop and eat. Crowded like crazy on the weekends, but easy enough if you take the train to get around. Pasadena is only about 20 minute car drive from the Conv Center, but still have to deal with traffic and parking. Parking, btw, is hit and miss if you have to pay for it or not.

Fry's is fun, but I'd avoid buying anything you aren't willing to take back. Why? Every hard drive I ever bought there needs a replacement or 2 before it works right. Don't get me wrong, it's great for prices, and everything BUT hard drives works great.

Beaches: Santa Monica, Malibu, Venice are all pretty close. Manhattan beach is (I think) a 10 min drive from LAX. Newport and Huntington are about 40 minutes away, but in Orange County and sometimes less crowded. Venice and Huntington are the better selections if you want to window shop for trinkets.

Disneyland, Knott's Berry Farm and Magic Mtn. amusement parts are all within easy driving distance.

Wineries, if you're into that, north into Santa Barbara (about 2 hours up 101), or down to the Temecula area (not very quick, but 10 East to 15 South will get you there). It's about 90 miles away.

Museums: Univ. Of Southern Cali. has a GREAT set of museums just minutes south of the conv center on the 110 fwy. Several types (natural, science, health, etc), for cheap or free. Get out before dark, cuz the neighborhood is horrible. The LaBrea Tar Pits are near the Groves (you might smell them on hot days), goodle it for what they have, but it's pretty amazing. Near the Tar Pits are a bunch of LA County museums, ie art, sculpture, etc. The Ghetty Museum is near UCLA, about 20 minutes or so from downtown. Also art oriented. There's the Botanical Gardens out by Pasadena - I forget their name, but the hotel should be able to tell you.
If you need more questions, or have questions when you get there, feel free to PM.