Does anyone have a readable map, on line , of the Rome city buses.

Rome buses are really convoluted. I would search instead for specific lines. Say you are going to the Borghese Gallery. Use Google maps to click on the Gallery then, click directions, you can choose a second destination pretty much anywhere on the map and if you switch from car view to walking or public transit you can see which routes serve the area. Doing this filters out all the "white noise" of crossing lines and you can clearly see where the line goes and if it truly gets close to where you want to go. Remember that many smaller streets are not served by buses so getting close may have to do.

Once you know which routes go to your destination try opening another google window, under the search box click on images, then enter your route number and the word Rome.
I've gotten you started here.

I also really like this website, it has tons of information on public transportation and even gives opinions on which to choose when there is more than one option.

Here is one you can print but you may have to print in sections to get it big enough to see. The google trick will help with planning but I usually print several and/or buy a map on arrival.

These maps are available two flights down the Metro Subway station right there in Piazza Cinquecento at the north end of Termini train station in Rome.

For quick reference check the Kiosk just outside the doors leading to Piazza Cinquecento for a list of which bus numbers go to popular tourist spots. And you can always ask the staff in offices at the platforms placed throughout the Piazza. Those platforms also display maps showing where to board bus numbers in the Piazza, which is huge! Also each platform displays bus numbers it serves!

Each platform also has a sign listing all the stops for each bus number, that's the time to find your stop and count the number of stops before it so you know when to get off! This doesn't work on bus routes with detours or reroutes tho! The driver or other passengers will have to help you there! Also you have to know the name of the stop you need for your particular destination. If Google doesn't help, ask other passengers or staff at the platform. Best to do this before you board! Or ask the staff at the Kiosk!

If you sit up front on the right side of the bus (passenger side), you can see signs listing stops for that bus number posted at each stop the bus makes on its route.

Following Sandra's advice, also note that some buses have stops other than the lanes with capital letters, alongside Stazione Termini, or at the boundaries of Piazza del Cinquecento.

Rome2rio.com may be of some help.

Thanks folks , you guys are really fast and very helpful.