Were there any families on the Enterprise-E like there were on the D? Was it also a ship of exploration?
No, as far as we know families were not kept onboard. The Sovereign-class vessel was designed with threats like The Borg in mind. Overall, it had a sleeker, more streamlined design, as well as the most advanced offensive capabilities in the fleet. It WAS still designed primarily for exploration, as that was the official purpose of Starfleet, but it was also built to have more "teeth" when things go wrong and hit the fan.
The design did not allow for spacious living quarters like its predecessor, with far less provision for civilian crew or visitors. Obviously, there were still opulent quarters provided for diplomatic visitors, but that was the extent of it. In a pinch, however, as with emergency evacuation situations, the ship could hold up to 6500 passengers.
As far as crew quarters, married couples were allowed to share larger quarters. Although it's never seen on-screen or mentioned off-screen, one would assume that a new baby would stay with the parents until such time as other arrangements could be made.
For more information about the Sovereign-class design, please see its article on Memory Alpha.
No. There weren't.
In an AOL Webchat, Ron Moore (The author of the Generation Screenplay) stated that, during the events of First Contact, there were no children aboard the Enterprise-E:
Q. Is it true that Enterprise E has no children aboard ?
A. This was definitely the case in FC, I don't know if Michael and Rick plan to change this in Trek 9.
Moving down the canon scale, in the Trek Novel Star Trek: The Next Generation : Q&A, there's a brief mention to the Enterprise-E being largely child-free but not totally:
“When you were raising Wes, they allowed families on ships." "They still do, to some extent," Beverly said. While the Enterprise-E did not accommodate the familial needs of large numbers of its complement the way the Enterprise-D did, families weren't completely forbidden, either. This was in keeping with Starfleet policy in the wake of the arrival of the Borg and later the Dominion. While there were decided advantages to not separating officers and crew from their families for extended periods, there were concomitant disadvantages to having children and civilians on a starship that faced dangers both predictable and unexpected on what seemed like a weekly basis.
Still, exceptions were made. Beverly's departed head nurse had raised her son Noah on the Enter-prise--E—and alone, after her husband died in the Dominion War. The eight-year-old Noah was now with Alyssa on Titan.