We are staying out by the Vatican and want to purchase a transit card for 72 hours, which is the most economical pass?
The ATAC site isn't as user-friendly (for me, anyway) as the previous version. Here's the info on the 72-hour pass:
https://www.atac.roma.it/en/tickets-and-passes/roma72h
If you are staying within walking distance of the Vatican, you may not need a 72-hour pass if spending a day there. We've also walked all over Rome and rarely needed to use public transit. Also, understand that it doesn't cover your transport to or from the airports. Here is the info on the 48-hour pass.
https://www.atac.roma.it/en/tickets-and-passes/roma48h
And the single-use BIT ticket, which is what we've used for the rare occasions we've wanted to use central Rome transit. I've purchased a few from a newsstand and just tucked them away until needed.