Public transport
 

Has anyone travelled by bus or train from Logan Airport Boston to Essex Massachusetts? Is it even possible and if so is it easy and what does it cost? Thanks for any information.

I haven't done it and it wouldn't be straightforward. Amtrak doesn't have a route that direction (they go more west of there).

There is a bus from Logan to Newburyport MA, which is only about 11 miles from Essex, you could take a taxi from there.
https://www.ridecj.com/locations/boston-logan-airport/

Otherwise, one of the MBTA rail lines goes to Rockport MA which is also about 11 miles from Essex. (I think they also have a line to Newburyport) But they don't have a rail station at the airport. So you'd have to get the line fro the airport to North Station to get the North Newburyport/Rockport lines.
http://www.mbta.com/schedules_and_maps/rail/

It looks like one of those MBTA lines stops in Ipswich, which is closer to Essex, only about 6 miles.

Where are you going in Essex? Take a look at the commuter rail map linked above. Manchester-by-the-Sea may be the closest stop.

Easiest would be to rent a car since you will need one there unless you're staying with friends.

Also look at the bus that runs from the West Gloucester Commuter Rail Station. It doesn't run frequently. http://www.canntran.com/PurpleLineEssex.cfm

The only way I figured out how to do it involved either a taxi or a bike for the last 11 miles as mentioned above. That would not have been practical for my group- for one thing, as mentioned, you need a car while there. Car rental turned out to be the cheapest, easiest option.