I took the airport express bus from the airport to/from Atocha. Only 5 euros each way. And it only stops in a couple of places so it doesn't take that much longer than a taxi, although you will have to walk to your final destination from Atocha (or take a taxi of course).

I didn't find the airport especially chaotic. But the walks are very long. I flew Delta from/to JFK and the USA gates are at the far end of the airport which made for quite a hike.

I wouldn't expect to do much the first evening. If you are coming from LAX I assume that's a fairly long flight. Unless you are impervious to jetlag.

For Sunday it really depends on what you like to do. It's a giant metropolis. I wasn't a huge fan of the Prado. But I went at the end of the day when entry is free so I had 2 hours to see things. It's lots of old painting mostly. The part I liked the most was the treasures of the Dauphin exhibit hidden in a vault in the basement. Lots of cool and interesting things in that colleciton.

I loved the National Archaeological Museum. It was closed for several years for renovation but only reopened in the last couple of years. Tons of artifacts from all of Spain's history well-displayed and interpreted in English as well Spanish. On Sunday it closes at 3 I think. It's only a couple of euros or free after noon (I think). The museum is about 25 minutes walking from Atocha.

Of course there's the Palacio Real which I thought was a bit overpriced. And no photos allowed inside. But it's a massive palace which is kind of cool. The gardens are nice too.

I agree that Puerta Sol and Plaza Mayor area are worth a wander. I liked Retiro park alot but it was very hot the day I was there so I didn't stay all that long. You could easily spend a lot of time there as it is enormous. But it's more of a relaxing, people-watching kind of place.

I stayed at the Eric Vokel apartments hotel and the Wootraveling apartments on Calle Atocha. Eric Vokel was nicer but also more expensive. Both are serviced apartment type places so they don't have all the hotel amenities but that makes them less expensive. Tons and tons of hotels in that area though.