Ciampino airport to termini station

1. Ciampino is only about 15 km from Rome, so a Taxi would probably be the easiest solution (you may need two Taxis for a group of six). There are also Shuttle Buses. Which airline are you using to Ciampino?
2. Venice is a bit of a distance for a day trip, but it's certainly possible. Using one of the high speed trains that travel at up to 300 km/h would be the best option. On the direct (no changes) trains from Roma Termini to Venezia Santa Lucia, travel time is 3H:45M, so if you leave early and come back late that will provide the better part of a day in Venice. If you're willing to commit to a specific train, date and departure time, you can save some money with advance purchase Super Economy or Economy tickets. The discounted tickets are usually non-changeable and non-refundable after purchase, so choose carefully before buying.
3. For the main attractions in Rome, I'd suggest packing along a copy of the RS Rome or Italy guidebooks. They have an enormous amount of information that will allow you to pick the sights that most interest your group, and help the trip to go smoothly.

1. Here's the information for shuttle buses from the airport to Termini:
   http://www.adr.it/en/web/aeroporti-di-roma-en-/pax-cia-bus
   I might suggest either SIT or Terravision: they are inexpensive, safe and comfortable.

2. Venice is really far away from Rome for a day trip: you'll spend close to 8 hours on trains. Personally? I would choose something else and save that one for when you can either stay there or a lot closer.

3. Which attractions in Rome you should see depends a lot on what your group will find interesting. A good place to start is with some guidebooks! Check them out of the library or find them at bookshops.

4. Most churches in Rome are free to visit as are the parks and piazzas. There are some Roman ruins which are ticket-free as well but again, they'll only be worthwhile if you find those interesting. :O)

The best free thing to do in Rome can be just to walk!
Where are you traveling from?

Really think Venice is out of the question. You will spend up to eight hours on the train and waiting for the train for maybe six hours in Venice. Hardly worth it. Any good guide book will list the top attractions in Rome. None are free.

Have you considered a day trip to Florence instead of Venice? It's only 90 minutes each way from Rome, so much more practical as a day trip.

From Ciampino airport there is a frequent cheap shuttle bus to the train station in Ciampino self (€1!) and saw at the airport also buses going to Termini in Rome. I stayed in 2012 in Ciampino and took the direct train to Termini in Rome and was also not so expensive, less as €4 if I remember well. A taxi mini van can be ofcourse more convenient, but will make it sense using it staying so close to Termini.
http://www.ciampino-airport.info/transportation.html