For discount shopping in LA proper I would suggest staying somewhere near the Beverly Center. In addition to the stores in the mall, across the street there is The Beverly Connection which has Nordstrom Rack, Loehmanns, and Marshalls (and I think a Ross too). You can also use taxis and buses to get to most other US retail chains without too much trouble or cost (though not too many Walmarts close by).

If you went to Woodbury Commons outside of New York, the Camarillo Premium Outlets mentioned by N1120A are your closest match (both part of Chelsea Premium Outlets Company). Here is a direct link to the transportation page of the Camarillo Premium Outlets. Two things mentioned on there in addition to Amtrak is taking the Ventura County Airporter from LAX to the Oxnard airport (which is not too far) or Roadrunner Shuttle who would take you as part of a tour. On the road runner site saw they do a wine tour with pick up/drop off from the Outlets - so perhaps you could make a whole day out of the trip out.

A little farther east of LA Chelsea Outlets also has a center called Desert Hills Premium Outlets. I only mention this because on their transportation page they link to a company called California Best Tours that seems to have an organized shopping tour (though in Japanese!). Did not see a similar tour group for Camarillo (though it probably exists).

If you go to either of these two register for their VIP club ahead of time and you can print out coupons for extra savings (the VIP discounts are slightly higher than the coupon book they hand out at the management office).

Ontario Mills is another outlet mall that is about the same distance. Similar to The Mills Outlets in other metropolitan areas - though these days the number of true outlets vs. regular stores is not as great as the premium outlets.

Your other question was about Anaheim. For Orange County the most centralized shopping would be staying closer to Costa Mesa and South Coast Plaza. From there you could also cab to another Simon outlet-ish property called The Block at Orange - it is as much about entertainment as shopping, but they do have a Neiman Marcus Last Call and Saks Off 5th (similar to Camarillo). Plenty of Targets, Walmarts, Marshalls, etc in Orange County as well - but those will be more spread out.

Also to add to jackal's comment - just survived driving on the wrong side of the road in England for a week - so would also put in a vote to try driving over here - especially if you stick to major roads and highways. After a day or two you'll adjust. Just be careful to not hit a curb when turning right and go slow! Plus with a car you can hit all these areas and then some.