Moscow (Russia) and Tallinn (Estonia) are 867 km (539 mi) apart in a straight line. The way we suggest is by train and bus, via Saint Petersburg and Ivangorod: ~9h 20m door to door with 2 changes. The fastest route, by train and bus via Tver, Saint Petersburg and Ivangorod, takes ~8h 35m. No flight is needed. The trip crosses one border, so check what papers you need.
Step by step
Train from Moscow to Saint Petersburg: about 4h, 635 km
Bus from Saint Petersburg to Ivangorod: about 2h, 134 km
Train from Ivangorod to Tallinn: ~2h 50m, 196 km
Every route we know
Train + Bus + Train, via Saint Petersburg and Ivangorod: ~9h 20m, 2 changes
Train + Bus + Train, via Tver, Saint Petersburg and Ivangorod: ~8h 35m, 3 changes
Flight + Bus, via Saint Petersburg: about 11h, 1 change
Flight, via Belgrade and Budapest: about 9h 50m, 2 changes
Flight + Train + Flight, via Minsk and Vilnius: about 9h 55m, 2 changes
Times are rough door-to-door times, with time to change and to get through airports included. A "~" marks a rough time worked out from the distance. We don't have timetables, and fares are rough estimates, not quotes: check Google Maps or the operator before you travel.