Sorry, I have to disagree with Lennythelion at #7 - booking one way transatlantic flights is the height of absurdity when the OP has made it clear that budget is a primary decider.
You might be able to find an open jaw ticket (a return flight into one city, and out of another) without a massive price increase, but if you're trying to minimise costs, then there is no chance that two one way tickets will be the cheapest way to travel from LAX.
