5 nights in Vegas? Are you planning to do an overnight trip to the Grand Canyon from there? If not, I would advise you to cut it to 3 days. Vegas hotels are cheapest Sun-Thurs.
It's possible to do LA by public transportation, but it's a lot more convenient by car because of how spread out it is and because some places are not easily reachable by public transportation.
I would recommend renting a car in LA.
Less than $100 per night is a pretty low budget for LA. Are you aware of Hotwire.com and Priceline.com? If you're willing to read up on them at www.betterbidding.com, you can get a deal for around that price in a decent area (Hollywood/West Hollywood). Basically, you get a hotel without knowing which one until after you pay, but you choose the star-level and neighborhood. There aren't really awful 3-star hotels. You can also get a private room at a hostel like HI in Santa Monica.
In Vegas, you can get hotels for cheap Sun-Thurs, so you can save some money there and add that to your hotel budget in LA. Check out http://www.Travelzoo.com for hotel deals as well.
You can drive from LA to Vegas or you can fly for about $50 (Kayak.com, Southwest.com). You can also take a Greyhound bus (this would involve some time getting to and from bus stations in both cities).
I would also recommend for you to add either Grand Canyon or San Diego to your trip, or both. San Diego is only 2 hours from LA, either by bus, car, or train.