The easiest way to get to Narva is by bus and then you'll get to this LITTLE bus station. It is located ~1km south-east of the city centre. The railway station is close to the bus station but since there is only one train a day in each direction to/from Tallinn, the train is no option. (The train leaving Narva departs ~05.30 or so in the morning.)
The bus journey from Tallinn will take roughly 3 to 4 hours and cost you from 9 to 11,40 Euro.
I put a link here to a good site where you can plan your journey.
When I went back to Tallinn we arrived 2 min ahead of schedule after a journey of 212km and with a lot of road works. IMPRESSING!!
Since this is the main road for getting to St Petersburg, there are more or less always lots of people and vehicles waiting to pass through the border control. Everyone has to get his/her passport and visa checked at both sides of the border which means it will take quite some time to get through.
I have understood that the procedures are a bit easier for Estonians, or at least for the people living in Narva, since they may perhaps cross the border several times a week.
You can take the bus no 31 (10 est.kr single trip) from central Narva, somewhere, and reach Narva Jöessu in about half an hour. Disembark anywhere at the place, and you can easily access the nice long beach of Narva Jöessu and walk there. There are some restaurants, and I ate at Lemmik which didn´t have all of the food on the menu, but not expensive at all.
I'm writing in red because I want to warn you! Never travel from Estonia to Russia with a car. We had to wait over 10 hours! First we stood on a huge parking lot, waiting for a number. The sun was shining, it had about 28 degree Celsius and it took over 6 hours. There was a hole in the ground surrounded by big walls, so that no air could circulate - that was the toilet. What a odor! I used it once and never again. After we got the number we had to wait for the Migration-Pass. Then we waited for...not sure what .... It was just a sheet of paper. Then we were waiting here, standing in line there..... all in all it took 10 hours.
There are some bus links from St.Petersburg to Tallinn (360 kms). But you can save some money if you'll take a bus from St.Pete to Ivangorod (60RUR or USD2). Then cross the border, turn to the left and board Estonian bus from Narva to Tallinn from bus station if you want. It costs EEK110 or USD7. ISIC saves 50%.
Or you can follow me and hitch-hike the bus ;-)