L.E.
I don't know what area you are going to be staying in but here are some recommendations based on different areas. I have lived in LA for 12 years and with toddlers for 4 years. I was a nanny before that so if you have any more questions about specific areas, you can message M.! I have taken kids as young as yours to disneyland, and it is do-able, but it is really more for you at this age.... to see your child smile and for you to see disneyland. There won't be many rides your son can go on but there is an area, toon town, he can wander all around without having to wait in long lines (which can be difficult at 19 months).
Other great ideas with kids: in Los Feliz in Griffith park there is a train ride and pony rides. Kids love the rides! There is also a travel town train area further into griffith. On saturdays and Sundays you can also ride the Carousel in Griffith park. Also, there is alot of hiking and exploring in this area.
During Thanksgiving it is fun to go to the Grove with small kids--- this is in the "middle of LA" pretty much, it's a big outdoor shopping mall off of Fairfax... I think Santa Claus shows up in his gorgeous house around thanksgiving time. Sometimes they have snow! It will be crowded though but you can ride a trolly and watch the fountain water show...
If you are further south or near long beach, the Long Beach Aquarium is an incredible place to go to with toddlers--- they can just run all around and look at all the fish. It's fun for adults too.
Something you won't find anywhere else is the Santa Monica Pier in Santa monica. There are rides and games and it's pretty much a fun spectacle for that age (mostly to people watch). He'd probably enjoy the Carousel and walking down the beach. There is an arcade along with plenty of food, the world famous ferris wheel and lots of rides...
Of course, the beach is a great place to go too, and I would suggest Manhattan or Hermosa Beach with small children. They tend to be the cleanest and have the easiest parking... Santa monica is nice too but it is a bit more crowded.
If you are in the valley area there is much to do in the way of parks and kids stores and that kind of thing. Lots of shopping for mama on Ventura and plenty of things for a toddler to look at (if he's in a stroller)...
In Hollywood you could go for a hike up runyon canyon with your son and "explore" --- that might be alot of fun for him! Or even take a stroll down Hollywood Blvd and shop/sight see at Hollywood and Highland. LOTS for a toddler to look at there but don't forget the stroller or carrier as it's very crowded on Hollywood Blvd.!
Have fun!