To ferry travel between Finland and Estonia you will need an ID card or passport. There are four ferry operators between Helsinki and Tallinn, which include Eckerö Line (cheapest and slowest), Linda Line (cheapest fastest ferry), Viking Line, and Tallink.
If you can try traveling there over the summer. During winter it can get frustrating because ferries get slowed down by ice and winds. Two ferry companies I used between the cities of Helsinki and Tallinn were Tallink Silja Line and Linda Line. Both offer several daily departures but I liked Tallink because they offer overnight cruises, so you won't have to book a hotel.
I've done both the ferry and plane back and forth between Finland and Estonia and overall the ferry is a better bet because the whole airport nightmare will be more of an inconvenience and still take you much longer because of checking in.