OOOOOOOO CHILD. Okay. The LA low-down. First of all, the triple threat of cheap, safe, and close to campus is nigh-impossible. Your best bet is to go through the USC off-campus housing that's available to graduate students. Even then, I know current grad students that have had to deal with break-ins and the like. It's the reality of LA, especially when you're downtown. That being said, the immediate areas around campus are a lot safer than they used to be.
Now, LA public transport. Buses. Just... don't. The Metrolink is okay, depending on where you're trying to go. Amtrak is okay, business class is lit on the Pacific Surfliner line, which helps if you want to visit Orange County or go down to San Diego. Did y'all know that they serve snacks and wine to business class in the afternoon?! Found that out when I went to San Diego Comic Con for work last year. 10/10 recommend.
If you ask any SoCal resident how long it is to get somewhere, the answer will almost always be "About 20 minutes". This is lie. Do not believe it. LA traffic is insane. I think the only place I've ever been that had it beat was downtown Seattle. Anyway. For example, I live approximately 25 miles from USC in north Orange County. During morning traffic, it would easily take me 90 minutes to get to campus... assuming no accidents. It took me 2 hours and 15 minutes to get to Westwood for my UCLA interview on a Thursday. I left at 8am and was still a little late for my 10am check-in time. So an extra 10 miles took an additional 45 minutes to drive.
With all that being said. because public transportation is such shit, you really can't be an LA local and not have a car, or at least access to a car. If you can't afford to buy one or don't want the hassle of owning your own car, it might be a good idea to find a roommate and share a car? Or, you can become a member of a car sharing service like ZipCar, which a lot of my friends did for undergrad.
Other things... Griffith Observatory is cool and worth the trip. LACMA is also very cool (and free general admission the second Tuesday of every month). Hollywood (the city) is a piece of crap. Just look at the sign and move on - it's falling apart and no one famous hangs out in Hollywood anymore, so it's really just a bunch of cracked stars on the ground and walkways that smell like pee. The famous people are all in Malibu and Calabasas and areas where it isn't crap. The Broad Museum is a little overrated and you only get like 30 seconds in the Infinity mirror room but it is kind of trippy. The best coffee is at Rubies + Diamonds. Don't go to bars/clubs that have cover fees, they aren't worth it when there are so many cool places that don't have a cover. But also be prepared to pay like $16 for a cocktail because... LA.
Other than that, my nightlife knowledge centers mostly around Anaheim because... I go to Disneyland a lot (fun tip that won't be relevant once Star Wars land opens: Tuesdays are the least crowded day!)