Our FAQs section (a link at the top of the TOPIC LIST) discusses this in detail. You'll need to get a TAP CARD (credit-card sized card upon which your fare will be 'loaded'). The FAQs has a list by zip (postal) code along with a map of the stores that have the card (although this info is a little bit dated).
To my knowledge, the TAP CARD isn't available at LAX (a pity). The closest places to LAX at which to buy the card (and associated WEEKLY PASS) include:
1. The Green Line METRO RAIL station adjacent to LAX (there's a free shuttle to it -- info is in the FAQs in the "Local Transportation' section).
2. The Ralphs in the commercial district (shopping area) to the north of LAX.
If you'd rather buy your TAP CARD near your hostel, the closest dealers are:
1. Freda Mohr Senior Center -- 330 N. Fairfax Av (just north of the northeast corner of Fairfax Av & Beverly Bl)
2. Honest Max Bargains -- 7909 Beverly Bl (northwest corner of Fairfax Av & Beverly Bl).
You might want to call ahead to confirm that these locations still sell the card.