One word about Super Shuttle:
It's one of those van services that cruises around the airport picking up fares and then drives to each location serially. I know some people who have used it and liked it. However, I know far more people who complained that, after getting picked up at the airport, they were driven round for 2 or 3 more circuits (and LAX is a fairly big airport in which to do that) until the driver had a full van, and then spent hours more getting driven around southern California until it was finally their "turn" to get dropped off. Once, when my elderly mother get re-routed from BUR to LAX due to weather, AA put her on the SuperShuttle to get to BUR -- it took more than 3 hours for what should have been a 45 minute ride at that time of night.
If your finances are really tight, you might consider SuperShuttle, though I would think the price advantage evaporates when you're talking about a family of four. When I lived in SoCal, I'd usually take a car service or a cab to the airport -- pricier, yes, but time is money, particularly in the OP's case because he's on vacation with a very limited stay in the LA area.
The OP should also note that LAX is in the southwest part of the city, not far from the county line. It is not at all close to Hollywood, downtown or Universal City because of the way the freeways are laid out in the basin. It is not feasible to use LAX as a transportation hub to get from LA to Anaheim if the OP is staying in Hollywood, Universal City or West LA.
Southern California is absolutely unique in that it has the worst public transporation system in the world. Virtually everyone drives -- for the most part, the only people who take public transportation are those that can't afford cars. Though it is possible to get around without a car by only using public transportation, it is not easy, pleasant or efficient.