So for my transportation to LAX I always do a one-way car rental from a non-airport Avis location. The rate is regularly around $40. I have to pick up the car the day before but some neighborhood locations will pick you up if you call them and ask. The one near me is close so I can usually find someone to drop me off. Once I have my car I can leave my house when I need to and then return the car to the LAX Avis location. Then I take their shuttle to the terminal. Super simple.
Getting home from the airport can be a little more tricky, but it is still cheaper than the shuttles. The problem is that if you rent a car from LAX you will have to pay a ton of airport taxes. I found that Budget is usually the cheapest, but still can be around $50. My newest find is an Avis location that is not the official LAX branch, but is next door to Dollar. So you take the Dollar rental bus from the terminal and then when you arrive at Dollar you walk about 100 yards to a "neighborhood" Avis location. Because it is considered a neighborhood location, you don't pay the ridiculous airport fees. I recently paid $41(the LAX Avis would've been $80) for a much more convenient and inexpensive ride home from LAX.