We just got in yesterday from a three week trip through Italy. Most of it via rental car. I rented a car through Auto Europe and picked it up from Rome Termini. To get to the A24 from the Termini without hitting a ZTL makes you take a few turns getting out of the neighborhood but it is possible. Here are the directions I got from another travel site. (It was cut & pasted into my travel file. I'm sorry I cannot find the direct link)
"Most rental car companies hold their cars in a parking structure located on via Giolitti. Because of the absurdity of the one-way street system in this part of town, you will need to circle around to get back to another section of via Giolitti in order to access the tunnel under the railroad tracks. Don't worry; it's easier than it sounds.
• Exit the garage and turn right onto via Giolitti.
• Go three blocks and turn right on via Cairoli.
• After one block turn left onto via Principe Umberto.
• Two blocks later turn left onto viale Manzoni.
• You are headed toward the station; turn left back onto via Giolitti.
• At the traffic light one block up, go right.
• This puts you onto via Tiburtina, and takes you through a tunnel under the railway tracks. Stay on via Tiburtina (through the wide intersection where five streets converge, Tiburtina is the stone, wider street in the center), continuing along to the "tangenziale" and following the green directional signs for the GRA and Autostrada.
• When the street merges with the tangenziale you need to be in the left-hand lane to access the motorway.
The via Tiburtina will lead you to the Grande Raccordo Annulare (GRA) which is the ring-road around Rome and from there you can access the various Autostradas (A1, A24, etc.) to carry you off to the gentler roads of the peaceful countryside."
I hope this helps.
We drove from Rome to Cinque Terre (stayed and parked in Monegila), on to Lake Garda and the Dolomites and turned it back in Venice.