From central/Downtown LA, you can take Venice all the way to the beach with a dedicated bike lane. I rarely go on trails, as I only have a road bike. My favorite trip is to take the bus to somewhere north of Malibu and ride along the PCH for about 20 miles.
Off-topic, but LA really does have great public transportation. It covers nearly the entire metro area such that you can get pretty much anywhere and only have to trek a few blocks to catch a connecting metro line (whether that be bus, rail, or subway). The problem is you're not going anywhere fast.
I never do trails either since all I have also is a road bike. There is a stretch of Mulholland, where it is a dirt road, that is technically passable with road tires, but I have yet to attempt it.
I like Ventura Blvd in the Valley to make it to the beach. It is congested at points earlier on (east), but then it clears out. Venice is busy almost the entire stretch.
Off-topic, but LA really does have great public transportation. It covers nearly the entire metro area such that you can get pretty much anywhere and only have to trek a few blocks to catch a connecting metro line (whether that be bus, rail, or subway). The problem is you're not going anywhere fast.