From now until October 5, the Cercanias between the airport and Chamartin is under renovation and has been replaced by a shuttle service between the affected stops, making this rather unwieldy, especially for a someone who has just arrived in Spain and has baggage. There is an express shuttle bus that goes from the airport to Atocha with a few stops in between for just 5 euros. This is the easiest way to get there via public transit. You can pay on board via contactless credit/debit card or cash.
Ignore what the poster above said about finding the bus being one of the most stressful things you'll do. All you have to do is exit the airport building on the arrivals level of your terminal and find the bus stop. It's not that deep or hard.