score:10
There is already a good answer, but here is some more detail.
Taxi would certainly be more convenient, but depending on your group size and luggage you may want to pre-book something that can accommodate your needs. From the sound of it, you may not fit into a regular taxi. You can try a ride-share like Uber XL. This would also eliminate the need to get from Mainz Hauptbahnhof to your final destination. On the other hand, it will be more expensive.
Upvote:10
Mainz Hauptbahnof is a 20min train ride direct from one of the airport train stations
This is the most straightforward and least expensive option, but, depending on the luggage amount and the kids' ability to hold/walk, it may be a bit difficult.
As it's only a 30 km ride, you can also take a taxi, this has the advantage of arriving directly at your destination, being able to carry all your bags (Beware that there might be a bag supplement). A site quotes €55 for a day ride from FRA to Mainz.