There's some decent shopping in Woodland Hills, a middle-class city in the southwest part of the San Fernando Valley. It's accessible via the Orange Line at Warner Center, a recent addition to Los Angeles's rapid transport system (well, it's Bus Rapid Transit, but it's a tad more user-friendly than a regular bus line).

There are several malls and a few hotels in the area. Here's a Google Maps overview:
http://maps.google.com/maps?f=q&sour...43945&t=h&z=16

And a Wikitravel overview (lists some of the shopping and hotels):
http://wikitravel.org/en/Woodland_Hills

The two main malls:
http://westfield.com/topanga/stores/
Wikipedia article: http://en.wikipedia.org/wiki/Westfield_Topanga
http://westfield.com/topanga/promenade/allstores
Wikipedia article:
http://en.wikipedia.org/wiki/Westfield_Promenade

Other stores nearby visible on Google Maps.

Burbank Town Center has OK (not great) shopping, but it's much closer to Downtown LA (less transit time) and is easily accessible from the Metrolink Downtown Burbank station:
http://www.burbanktowncenter.com/directory

Google Maps:
http://maps.google.com/maps?client=s...43945&t=h&z=16

Those (plus the Sherman Oaks Galleria/Westfield Fashion Square and Glendale Galleria, but those are harder to get to by transit) are what I'm familiar with, but there's tons more in L.A. This site has a pretty comprehensive list:

http://www.latourist.com/index.php?p...g-center-links

Search for those on Google Maps and combine that with searching for directions via public transit on Google Maps to see which ones are easier accessed.

The only outlet mall in L.A. proper is Citadel, which isn't right on any rail lines. It requires a couple of bus transfers, though there are a couple of hotels in the neighborhood (a Doubletree?). Otherwise, you'll have to get a good bit south, east, or north to find more outlets.

Details here:
http://www.latourist.com/index.php?p...-links-outlets

I assume you've looked into this, but is there a particular reason why you can't drive in the U.S.? Foreign nationals are perfectly fine to drive in the U.S. (with an easily-obtained International Driving Permit, if needed). Rental car companies are fine with foreign drivers, though you may need to purchase the damage waiver and liability insurance to be covered. Having the ability to drive in L.A. (which is NOT scary at all--big, wide streets that are almost always in a grid) will GREATLY expand your options (and allow you to carry around all the stuff you'll buy!).