Agnes is right. I've never come across a hotel that won't store luggage before check-in time. A few charge for this, but most do it for free. Note that apartments are another story; it's common that an apartment rental will NOT have a way to store luggage before check-in, although this varies.
Also, while they will never guarantee that you can check in before the official check-in time, if a room is available early, most hotels will let you check in early. Again, a very few charge extra for this, but most don't.
So, I agree with going right to your hotel. If the room is ready, check in, and leave your bags in the room. If it is not ready, leave your bags at the front desk (be prepared to remove what you need for the day, such as maps, umbrella, etc).
For either you or your friends, there are multiple methods of getting from the airport to central Madrid. While many are fans of the bus, metro, or train, I took a taxi. It's only 30 euros, fixed price, and no tipping is needed beyond this. So it's a good deal for a group, and if someone is tired or unsure or schlepping big bags, it's worth it even for a solo traveler.