Tallinn - Helsinki by ship (you get a discount)

Helsinki - Trondheim (several changes needed)
Helsinki - Ferry - Stockholm - Oslo - Trondheim
Helsinki - Ferry - Stockholm - Östersund - Trondheim :)

There are always 2 very much competing ferryboats HEL-STO, they also both do ferries from Turku to STO. The HEL ferry is very luxurious cruiseliner, full of drinking Scandinavians but still quite cheap-for ´deck or sleeperette=airline chair´ space.

You can-if you have time and are adventurous-go by train via the north_ overnight to Kemi/Maybe Tornea-walk/bus to border town Haparanda, then a few trains or bus to North Sweden-Lulea/Boden, then either south via train and change f.e. Gävle to go Trondheim or to Narvik in NO-maybe make detour via Lofoten islands-bus to Fauske, then train (by day-some nice views or overnite)-to Trondheim.

If you use the fairly straightforward route via STO you leave day 1 and arrive day 3.