Los Angeles is a very sprawling city with many different landscapes and cultures. There are many amusement parks, tours, hiking trails, beaches, restaurants, and an abundance of nightlife entertainment to fit anyone and everyone. However, keep in mind, Los Angeles is vast and not everything is in one area. Please research and plan accordingly as 30 miles away can mean a 3 hour drive.
The best option for traveling through Los Angeles is renting your own car. Luckily, it is a large city and has great deals on rentals. There is also Uber and Lyft, though this can get pricey as rates can be very high during surge times. I do not recommend taking the Metro as it is very outdated and many times you have to take multiple busses and trains to get where your going, which can be very confusing when you don't know where you're going.
This depends on what you're wanting to do. Different areas offer different vacation options. Do some research and decide what you're wanting to do. Below I have listed some of the most popular areas: Hollywood: Not a great place to stay. If you do choose to stay in Hollywood I would spend a little more money for a nice hotel. Most of the nicer hotels are in better areas of Hollywood. I also recommend seeing a show at the Pantages Theater if you stay near Hollywood. Santa Monica: Public beach life and shopping Beverly Hills: Fancy Shopping and people watching. Plus many mansion tours and celebrity siting's. Malibu: Private beach life (Quieter than Santa Monica) and laid back bars/restaurants. Orange County: Disneyland... That's basically it. Only go here if that's literally what you plan to do.