Best website to buy train tickets

A lot of people complain that the Renfe website can be finicky. I've used Trainline extensively for purchasing tickets and never had a problem.

Train airport to downtown is just local transport. Just buy at the station from a ticket vending machine. Madrid to Toledo is also just a (fast) commuter train really.

Trainline is a good site, as in Spain there are mutliple operators on some routes, and Trainline will show all of them.

Also when we arrive in Madrid, is it pretty easy getting tickets and a train to downtown?

The Circanias, or commuter trains, are easy if you arrive in Terminal 4 (Iberia, American, British, and some other, minor carriers.) The station is in the terminal. Those trains can take you to Atocha, but to get to Sol, the heart of the city center, you'd have to change trains (same system, no added cost.) You don't even need a pre-purchased ticket for these—you can enter by tapping a credit card (only on a limited number of turnstiles per entrance/exit) It comes out a 25-50 cents cheaper, too.

The metro is an option into the city from all terminals, but most city center destinations will require two transfers.

Thank you everyone! This is very helpful. We are staying a five minute walk from the Prado. I think this means we could take the Circanias that was mentioned to Atouche, then walk for 15 minutes or so?

You can get off that same Circanias train at the stop before Atocha, at Recoletos station. I'd map out which of the two was closer to your hotel.