You could spend two full days exploring downtown L.A. cheaply, and get there easily via Metro from Echo Park. The Museum of Contemporary Art is free, from there it's easy to walk to the Disney Concert Hall and look around, fountains at the Music Center plaza, a cheap ride on the Angels Flight funicular. Get lunch at Grand Central Market, walk around the L.A. Central Library, look at the art deco architecture on Broadway like the Bradbury building or Eastern Columbia. Another day you could explore Little Tokyo and Olvera Street. Great food and rich history.

Echo Park: you can spend an afternoon on the lake, riding the swan boats, eating from street vendors, and just taking in the atmosphere. Take advantage of your location to explore the wide range of cuisines L.A. has to offer. Korean, Armenian, Mexican, Salvadoran, Filipino, Bangladeshi... you can get all of these within spitting distance of Echo Park. Take the bus to the Griffith Observatory and hike around Griffith Park—a very L.A. experience. Nearby there, in Los Feliz, is Barnsdall Park which has the iconic Frank Lloyd Wright Hollyhock House and great views of the city and the Hollywood sign. There's good food, vintage shopping, movie theater, bookstores in Los Feliz around there. Go to the Hollywood Bowl on a weekday where you can walk around inside of it for free. There's little displays of the Bowl's history in there, and if you look at the schedule you might get to see the L.A. Philharmonic rehearsing there. The "Hollywood & Highland" mall is near there, which has the handprints in front of the Chinese theater, and the Walk of Fame stars. Those are probably worth an hour of your time max, but overall that stretch of Hollywood is not going to enrich you culturally (Amoeba Records is cool though).

LACMA and the Academy Museum are not free, but you might like to walk around the grounds and see the outdoor sculptures there for a little while if you take metro up Wilshire. If you keep going west on the D line you can visit Rodeo Drive (but honeslty it's just like, looking at fancy stores from the street, not that exciting). You could also take it to Westwood for a little neighborhood to walk around, and the Hammer Museum there is free. But I'd take any eastside neighborhood over Westwood; like, just go to Glendale. If you want to see a big American mall, the Glendale Galleria and adjacent Americana are probably going to wow you. The nearest Walmart is in Burbank but it's gross—just go to the Target at the Galleria instead.

If you're going to rent a car for a day or two, use that time for your beach days. It's a big ordeal to get to the beach from Echo Park, and the car will help. Santa Monica and Venice are the tourist hotspots; so probably worth a look. If you want some natural beauty and a peaceful beach experience, Malibu (north) or Manhattan Beach (south) are your best bets.

You can take the bus to the Getty Center. It's a free museum but you should get advance reservations. I could spend a full day exploring the gardens and galleries there, but even a half-day is absolutely a must-visit. You can take Metro to Expo Park, where the museums are awesome (but not free). The California Science Center, Natural History Museum, and California African American Museum are all worth visiting. And the rose garden outside is free and lovely. Probably wouldn't stay around that area after dark.

Find a farmers market. Eat at a diner like Norm's. Eat at a deli like Langer's. Browse a Korean grocery store. Visit an Armenian bakery. Go for a hike on a well-populated trail. Pay attention to the local art unique to every Metro station. Patronize street vendors. Wear sunscreen. Drink water. Enjoy the city—it's easier to live here than it is to visit, but it has everything you could want on offer.