I travel to/from Madrid quite a bit so I might have some advice. First, Madrid airport has 2 separate metro stops. 1 stop is for terminal 1,2,3 (which is where aerlingus fly from). the other stop is for the new terminal T4 with a stop between (called Barajas) both stops for the terminals. I assume that you will be flying home from T1 but just make sure if you return from the city centre by metro which terminal you need.
Either way get line 8 towards Nuevos Ministerios and I would recommend changing here to line 10 (towards Puerta del Sur) and get off at Plaza de Espana (4 stops). This is the main tourist area and you have the plaza, look for the Gran Via which is the main street starting at this metro stop, head towards Plaza Mayor, Puerta del Sol, also not far by walking to the cathedral, Royal Palace and Opera. I was there a few days ago and it is HOT in the city centre. You could get a metro at Sol or Opera which will bring you back to line 10 and back to the airport.
The metros are quick (about 25 mins) to the centre and go very often. I travel from terminal 1 and there is a good metro information desk at the entrance to the metro. You can ask here what is the best ticket for you and your family. the only drawback with the metro is that you have to pay a 1 euro supplement each time (going and coming) you use the airport stops. Don't ask why but is not a nice way to treat tourists. Also, be careful with your bags and wallets on the streets and the metro.
There is a manned luggage storage place in T1,2,3 between the metro and the terminal if you should need it. Not sure about T4.
On the return, make sure what terminal and if you need to use T4 or T4S , leave yourself plenty of time as these are very large with long walks. also if you are using the escalators on the metros or airport, stand on the right hand side, the left side is used by people running or in a hurry. I also find aerlingus check-in is also busy with long queues. hope this helps.