Sasso gives great advice.
Morristown is lovely, quite a bustling downtown and some good museums/historical sites. But the town is pretty suburban, in that you drive pretty much everywhere. I have cousins that live there (were born & raised there) and love it but we decided it was too spread out for us - I have an aversion to suburban sprawl (DH is from South Jersey so I knew what I wanted to avoid in terms of living here!). We moved from London last year and DH works in the city so the length of the train commute was also a factor.
We currently live in Maplewood and are in the process of buying in South Orange (town right next door - they share a school system and are v much sister towns) and love it. It's walkable, super friendly, liberal and diverse. The public/state schools are excellent. There are plenty of Brits/Europeans (I'm Irish & didn't want my kids to be the only ones with a "foreign" parent!).
For info check our any local websites, eg Maplewoodonline - this is a tremendous resource - people are v helpful, eg if you asked questions about living in Maplewood/Millburn and working in Morristown in terms of commute you'd get some useful info. It also covers Millburn, South Orange, West Orange and there are plenty of people on there who live in Chatham & Summit too.
For property, check out GSMLS - this is particularly good for if you're looking to buy. There is WAY more info than if you were looking at rightmove or the like!
Also check out Walkscore - it's not perfect but it gives a good insight into the walkability of potential addresses (this was a big factor for us - as I'm home with the kids and didn't want to be stuck in the middle of nowhere! - so was a column in the spreadsheet when we were looking at rental properties last Sept).
For schools, also check out Schooldigger.
And ask the relocation company for any info they have too.
Good luck with it all! NJ is a lovely state and we're very happy with the move.