Depending on what type of things you enjoy there is Knotts Berry Farm not far from Disney. There is a bus to there by Anaheim Resort Transport (ART)
Tour companies go to various places like Universal, trips around LA, San Deigo Zoo or Safari Park, Legoland etc..
There are also rail stations at Fullerton & Anaheim that you can catch Amtrak or Metro services to San diego or Downtown LA, among other places.
Newport Beach, Huntington Beach & Long Beach are not to far away depending on traffic but take a long time to get to on Public Transport.
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They already have days allocated in LA for doing universal etc .
Personally I think not having a car assuming you can drive is a big mistake . To do all the places the OP wants will be far easier with a car and probably cheaper .