Getting to Fiumicino aka FCO
Like most things in life traveling to or from the airport comes with choices. Without a doubt the easiest and in this case the most expensive is to take a cab or private transfer. With in the old walls of Rome a fixed rate of €48 applies. For a single person or a couple the cost may be higher then you want to pay. For a larger group it can make quite a bit of sense.
If on the other hand you want something more economical you have two very good options. From Termini station various bus lines go to Fiumicino for €6 per person. The obvious problem is if you aren't staying near Termini is you'll need to get from the station to your lodging.
The other choice is the regional train. The regional train heads to Tiburtina train station. The cost is €8. Tiburtina is a smaller and less hectic station then Termini. It's also across the street from the Tiburtina bus station. If you intend to take the bus out of Rome this may make your decision easier. But Tiburtina is further from the touristy parts of the city. Finally the regional train stops at various points along the route. Any of those stops may be closer to your lodging.
There is a third choice . The Leonardo Express for €14 will also take you to Termini. Obviously it's more then twice the price of the bus that takes you to the same station. It does have the advantage of free tickets for children under four years of age. Children under the age of twelve are also free if an adult buys a ticket. That means a couple with two children under the age of twelve could take the express train for a total of €28.
Getting to Ciampino
Ciampino is simpler . At least when it comes to choices.
The taxi rate from Ciampino is only €30 . A group of four should almost always make this choice if the lodging is with in the walls.
The bus choice is €5.
Unfortunately there is no train station at Ciampino airport. That means taking the train requires a transfer. It's not the greatest hardship but taking the bus is so simple that it's hard to suggest this choice.