Hi I went to LA last month and stayed in a hotel in Downtown LA, wasn't really near much as LA is such a big place everything is so spaced out! However we did Universal and Hollywood and both was accessed using the Metro which is their version of the underground! Hollywood was around 5 stops away and Univeral around 8 stops, quite cheap as well, $1.50 each way or $6 for a day trip if using it majority of day!
I would recommend to do a city tour, that way you can jot down the places you liked and go back!!

Have you tried http://www.vrbo.com Rentals by the owners...

If you want to stay in Hollywood I can recommend the Hollywood Orchid Suites. We've stayed there twice - it's not fancy but it's clean and has free parking (which can only be accessed by guests). It's located behind the Kodak Theater and is a couple of minutes away from the Hollywood and Highland metro station. They have one and two bed suites which both have fully fitted kitchens.