The cheapest way to get to Pasadena from LAX is to take the Flyaway shuttle and Metro Gold Line (light rail) combination. It takes ~80 min and costs $11.25 each way.
From LAX, follow signs to "Ground Transportation", to an outdoor waiting area. Cross to the median. Stand under the green sign, which reads "Flyaway, Buses, & Long Distance Vans".
The Flyaway Shuttle to Union Station comes every 20-30 min. depending on time of day (detailed schedule available here). Make sure you get the bus to "Union Station". The Flyaway takes the freeway carpool lane to Union Station, nonstop, duration 25-40 min. When you arrive, buy your $9.75 ticket at the kiosk and pick up your luggage. Note: the kiosk only takes credit cards, no cash.
Walk down the bus plaza and into Union Station. Follow "Gold Line" signs. Use the automated teller to buy a TAP card and fill it with at least $3 ($1.50 each way). Tap your card on the green circle to deduct the fare. There is no paper ticket. Take the Northbound Gold Line train, which should be signed for APU/Citrus College. You'll be on the train for 22 min. Get off at Lake Station. Climb the stairs, and walk uphill on Lake St. Take a left onto Santa Barbara St., walk 50 yards, and you're at our front door. Alternatively, walk to your hotel from Lake Station.
Schedules: The LAX Flyaway bus runs every half-hour, except for hourly service 1am-5am, and extra buses in the morning. Click "Union Station" for the current schedule. The Gold Line serves Pasadena until ~midnight, with trains every ~15 min (10 min in commuter hours.) Here are links for the Gold Line map and schedule.