Where exactly does the beginning of the game start. I always forget the route that the carriage takes when going to Helgen.
all I know is its on a hill in a snowy part of Falkreath.

Go from the gate inside of Helgen where you were about to be executed to the gate and follow the road.

It starts on the winding road south of Helgen and east of Greywater Grotto.

Isn't it also near the Falkreath SC Camp?

About the same distance Embershard Mine is from Riverwood.

that gate which leads to cyrodiil? i guess?

If you zoom in to the http://www.uesp.net/maps/srmap/srmap.shtml at UESP you'll see that the road out of Helgen forks outside the gates. One road goes directly east to Falkreath Stormcloak camp, the other turns southeast. This southeasterly road forks a second time, with one road heading toward Fort Neugrad and the other heading toward Greywater Grotto. These two roads connect southeast of Greywater Grotto. It is at this spot that the carriage ride begins.

The gate is a lie, there is a mystical force there that prevents you from leaving, and whispers things in your head, like "you can't go that way".

thanks.