You don't have to pay to get off the boat. If this is your first time, and you aren't familiar with dealing with travel arrangements, a travel agent is a good idea. They don't charge you extra, but often can give you perks like ship credits. You can get an all inclusive rate that takes care of port fees, taxes, and gratuities. Then you need to make decisions based on your drinking habits. Drinks are $6-$9 depending on what you drink. If you drink a lot, you are much better of getting drink package. You can check them out on the cruise line web sites.
June isn't a great time to go out of those ports. There are some Royal Caribbean cruises out of Bayonne that look good. The trouble with cruises out of NY and NJ is it takes so long to get anywhere.
I have done Bermuda three times, and I really enjoyed it. We did a Celebrity cruise out of NJ that was fantastic. I never had any trouble finding things to do in Bermuda. I'm not a big history buff, but I really enjoyed the history of Bermuda.
I did my last one out of Puerto Rico and it was nice being at a port the next day. Carnival isn't bad, if you have a load of kids going, it might not be a bad choice. It's a bit too party-party for me. And I did one Norwegian, it was out of Baltimore and it sort of sucked.
Anyway, go on expedia and sort out what cruises interest you. Read the reviews, then go on cruisecritic.com and read more reviews. Different ships and cruises cater to different people. Once you get an idea of what you want, then contact a travel agent. I know people who swear by cruisecompete.com. You put in the cruise you want, and agents make various offers. I've never used them as I do my own booking, but as I said, I know people who swear by them.
Excursions through the ship are easy, you just go on the cruise line web page and look for what you like. I prefer booking them myself as more of the money goes to the local economy. I'll like research, so I'll literally spend our of tripadvisor and cruiscritic, come up with some ideas, let my family have a look and we decide what we want to do. Sometimes, we don't plan anything, and just get of the ship and see what we can see. We went by cab to a secluded beach at Antigua that was fantastic. It had a little beach bar, jet ski and chair rentals, and not much else. It was beautiful and not at all crowded.
You may also want to consider Baltimore as a departure port, depending on what you chose.
If you have kids going, a cruise is a great vacation choice. They have programs that the kids actually enjoy, so you can have a nice time yourself and not your entire vacation wondering where they are or making sure they are entertained. Guilt free.