What is the best way to fit Bari into travel plans that include Venice, Cinque Terre, Rome? Easiest train route? Thanks
Well, it is well south of Rome, so go from Rome. 7 to 8 hours with one ore more changes by train. That being said, there is a Freccia which runs down the east coast, skipping Rome, taking 7 or so hours from Bologna so including the run from Venezia it would not be all that much slower from Venezia. Cinque Terre is completely on the wrong side for that trip.
Actually Bari is not that far from Rome. About 460 km (290 miles) to the south east. It's all freeway driving and, if you are not too shy on the accelerator, you can drive there in about 4 hours.
By train from Rome it's also just about 4 hours with the fast Freccia Argento trains (no train changes required). There are currently 3 such fast trains departing daily from Rome to Bari. At 08:45, 14:45 and 17:45 (that may change in summer, I don't know). You can certainly visit all those locations, the issue is how much time you have because you are all over the place along the Italian peninsula. Although it may not look that way on a map, however the Italian Boot is very long. Over 800 miles from the Brenner pass all the way to Cape Passero at the southernmost tip of Sicily. That is exactly as long as the Golden State of California from the Oregon Border to the Mexican Border.