Congratulations on entering the wonderful world of composting! Too bad about all those leaves, but now you know that never again will you send rich organic material full of garden nutrients via truck to the landfill!
I'm a keep it simple kind of gardener. To me it seems pretty obvious, you already bought a compost tumbler and it is sitting there waiting, use it!
The advantages of worm composting over regular (tumbler or pile) composting are:
1) you can do it indoors and year round. That's a help for us cold winter folks, but for you in LA, really doesn't matter.
2) it does produce a very rich, concentrated soil amendment. This is nice but if you do regular composting, you can turn some of your compost into compost tea (browse around in the monstrous long ACT [activated compost tea] thread we have going on) and get some concentration/ enrichment of your compost and turn it in to liquid.
3) you can put kitchen scraps in the worm bin directly, without anything (except some bedding for the worms) to balance it out. In your compost pile/ tumbler the kitchen scraps need to be balanced with some more carbon-heavy "brown" materials, like those leaves.
But while you can use some dried grass clippings as bedding in a worm bin, the worms won't really eat the grass.
So your problem if you did ignore the tumbler you bought and move on to worm composting, is that you still would need to do something with all the grass clippings. If you have a mulching mower that chops them up fine (most new mowers are) you can just leave the clippings on the grass. Otherwise you can use the grass clippings for mulch on flower/ veggie beds.
Your problem if you want to use the tumbler is that you have kitchen scraps and grass clippings that are both "greens" (if you aren't familiar with this read the sticky about greens and browns https://www.helpfulgardener.com/forum/viewtopic.php?t=9089). For them to compost well, they need to be balanced with some browns. Since you don't have the leaves any more, other browns might be shredded newspaper or other shredded paper (like from an office shredder), torn up brown paper grocery bags, cardboard scraps (never throw out those pizza boxes!
), etc. The greens and browns sticky will give you lots more ideas about browns.
The other thing about your tumbler is that it is for batch composting. That is you collect your green and brown ingredients then you load up the tumbler (not too full, you want things to be able to tumble around in there freely) and then you tumble it every day or so and wait for a few weeks. While you are tumbling and waiting any new compostables that come along have to be stored somewhere else. You can't keep adding to the batch in the tumbler, or else you will never be able to sort out your nice finished compost from all the newer nasty icky stuff you just added. So you have to have a way to store compostables for a few weeks, while one batch is working and you are collecting the next batch.
Hope this helps you think your choices through... As people have been saying do what works for you, so that it is manageable and you can stick with it. Don't over-complicate!