I'm a local.
My recs:
- Arts District downtown - Container Yard, EightyTwo (arcade for 21+ bar), murals
- Little Tokyo downtown - shopping, ramen/ sushi, roti bread at Cafe Dulce is great, Far Bar for drinks
- Main St. Santa Monica - boutique shopping, coffee shops, not as commercial as 3rd St. Promenade in Santa Monica where all the tourists go - Dogtown Coffee, Cafe Cielo are some favorites
- Sunset Strip - barhopping, live music shows - Roxy, Saddle Ranch, Whisky a Go Go, Viper Room
- Venice - Muscle Beach, boardwalk for cheap beach clothes/ sunglasses, oceanfront cafes
- Venice - Abbot Kinney - hipster/ artsy area of Venice - boutiques, Gjelina for pizza
The metro from downtown LA to 3rd St. Promenade in Santa Monica is also finished, so you can hop on the metro and take it all the way to the beach.