Lahemaa National Park
Lahemaa National Park is about 50km east of Tallinn and is a place to find some nice forests, seaside and swamps and bogs. One of the most suggested place to go there is Viru raba (Viru bog), that has 5km foottrack and watching tower. More info:
Many forests, swamps and bogs. Sea-side. Several pictures of forest and seaside area
Flora and fauna
The park is large and has many interesting places to go. Therefore it is hard to give one and only direction getting there.
By car. Drive from Tallinn towards Narva via Peterburi road. About 50km from Tallinn, there is a sign showing to Loksa (left). You may want to turn there as right after just taking about few hundred meters via this smaller road towards Loksa (direction north) there is the start point to Viru bog on Your left. Or You may want to drive to Loksa and further on to the seaside.
Better to buy a map from any petrol station in Tallinn before You go.
There are no entry fees, all the trails are free to hike. You should just remember that take nothing but photos and memories and leave nothing but footprints.
Viru raba (Viru bog) has 5km foottrack and watching tower.
You can also start and finish in same location if You go to tower and back or take a round trip back to start around the bog. There are good maps and information tables at the track. The track over the bog is made of wood and usually You do not need to worry about any special footwear. Sometimes there may be more water in the bog and waterproof hiking boots are recommended.
Hiking There is a number of marked trails in the park
You can put You tent free of charge to official "RMK" camping sites. Also there is nothing too wrong to camp outside the special site, but remember not to do any fire in a place that is not meant for making a fire. Putting up a tent on a private land is not allowed, You should ask permission from owner.
In a warm summer night You may want to experience the feeling of sleeping in the watching tower in Viru Bog. The tower has a roof, so no worries about the rain. Some times there may be mosquitos, that carry no disease here but are just annoying. Sunsets and sunrises in the bog are unforgetable! 
In forest there is a great possibility to catch a tick who can carry a disease (Tick-Borne Encephalitis or ehrlichia or borreliosis). Therefore a sufficient repellent is highly suggested. Also it is recommended to wear long light clothes and examine body in the evening to early find any ticks.
The waterholes in bog may be tricky to get out and even dangerous, so better not step off the track.