Best shuttle transport from Rome airport??

Romecabs is good. You can arrange/reserve online.

Are they reasonable? We have 6 people and luggage!

Don't sweat transportation in Rome. The cabs are easy to get. When you collect your luggage - just walk to the nearest cab stand outside. Have your hotel address/name written on a piece of paper. There are shuttle vans. The worker at the cab stand will then call over a cab for you. There is little cost difference between cab services. I find you get what you get, based on what's available at the moment. Same for going to train station. Have the hotel receptionist set-up a cab pick up for you the day before you leave. Or again, walk to the nearest cab stand and you will be assigned a vehicle based on number of passengers. The cabbies rotate who gets what customer at the cab stands. You may have to split into two cabs. Plan on a flat rate of 50€ for a standard size cab to get from airport or more for a vehicle for 6. Plan on 12€ for a standard size cab to get to train station from Vatican area or more for a larger vehicle. It's easy - don't worry.
We were placed in a van with another couple. The driver charged us as two separate groups. There were 5 of us total with room for more. You will just have to pay the total fee as one group for the van.

We are staying in a private apartment while there so there won't be a receptionist to arrange for us. The owner does not arrange either so we will have to do on our own. This is mostly my concern as we do not know the best way for this.

You might want to consider a private car service. Having a private driver meet you inside the terminal and take you to your apartment hotel will, in my opinion, be money well spent. No waiting in the regular taxi line, and no having a "lost" feeling. We have been very pleased using Rome Cabs. It is a private car service (not a cab company) that usually cost only a little more than a regular taxi. We always use them and have always had safe, courteous, and informative drivers. Go to romecabs.com and see what you think. We send them an email and they answer with a quote within 24 hours. Unless things have changed since our last trip you do have to pay cash (Euros) at the end of the journey but nothing in advance.

If your apartment is near the Vatican - takes notes of where a cab stand is while you are out walking around. That way, you can find a cab when you need one. There are cab stands at the airport, train stations, and near various major sites in Rome. (Example: Largo Argentina has a cab stand and is near Pantheon). If you go to Borghese Art Gallery, there are cab.s waiting on the road, just outside the gates of Borghese. I know planning can be a bit nerve-wrecking. It does help to have arrangements. Just know you have options available in case something doesn't work out. Enjoy your trip!