First Stop Madrid - Advice Needed

Carenir,
You can take the Cercanías C 1 Line (commuter train) from the airport (Terminal 4) to the Puerta de Atocha train station.

There are secured lockers at the Atocha station where you can store your luggage.
You'll find the "Consigna" (left luggage) marked in green (#5 on diagram) on the Planta 0 (ground floor) past the Tropical Garden (check out the small turtles!) and by Burger King (#37 pink). There's a security guard at the entrance of the left luggage place and your luggage will have to go through an x-ray machine prior to entering.

The Atocha station is HUGE and you'll find many shops and restaurants there but it may be quite overwhelming especially when one is suffering from jet-lag.

I would not do well going to a museum on day of arrival after a transatlantic flight.
It's been a number of years since I've ventured out into Madrid; on my last trip I only passed through (stayed in Toledo instead), so I'm sure others will give you ideas as to how to keep you busy and awake during those hours prior to going to your Airbnb.

Have a wonderful trip!

I took the bus from the airport to Atocha Station in 2016. My hotel was also in that area. I figured it would be easier with my too-heavy suitcase.

However you arrive at Atocha, it can be difficult to figure out exactly where you are (and which direction you're facing) when you step outside. Use of an electronic map with a you-are-here dot is highly recommended.

I opted to spend part of my Day 1 time at the Botanical Garden, which is quite near Atocha. There's a modest entry fee. The nearby and huge Retiro Park is free. Another possibility would be to do Rick's walking tour. If you're energetic, you can walk to the starting point (very rough guess: 30 minutes, but it can be figured out by using Google Maps). Otherwise, the Metro will work. That walk takes you to San Miguel Market, I believe, which has tapas and other foods available pretty much all day; handy if you're hungry at an odd-for-Spain time.

In May after an overnight flight we took the direct bus from airport to Atocha (€5), about 35 minutes. I also contacted our AirBnB host (we were staying on Calle Atocha) if we could drop our bags at the apartment early (around 11 am). He graciously allowed that, and even met us and gave us some restaurant suggestions. Worked out great for us.

Thanks all. Unfortunately, we don't have an option to check in earlier with our Airbnb host. She's booked solid with not much flexibility.
Anyone familiar with the Atocha-Retiro area? An option may be just to sit with our luggage, enjoy a mid-morning leisurely snack, with and take in the sites. Any place exist within walking distance?

I know it is possible to store luggage at Atocha - but I have no personal knowledge. Hopefully, it works out for you.

It's an easy walk to Retiro from Atocha. When we were there in October 2017, I was struck by the absence of homeless people, even in the more secluded areas. I didn't go out looking for them, but it was a striking contrast to the urban parks I've seen in the States.

Check your bags at Atocha. The fee will almost certainly be per bag, so it would be smart to do some consolidating if your group will be traveling with numerous small personal items.

Thanks, everyone. Feedback noted - why rush? We will check bags and it will give us the flexibility to explore upon arrival.