I started a new character two days ago and although I already have dozens of characters, on most playthroughs I still go to Riverwoord, Bleak falls Barrows and Embershard mine. The locations are pretty nice, and I always gain a few levels in the process and make some easy money. After that I go my own way.

So what do you do at the start of every new playthrough?

sneak my way to the lover's stone before going to riverwood

This seems to be the route of most of ny New characters I wander there absent mindedly sometimes with new characters.

In Oblivion I always used to swim across the lake the Vilverin and kill the bandits outside, then head to the Imperial City to sell my stuff, but in Skyrim I have no start of the game traditions.

Head off to Riverwood.

Some dudes were just about to lop my head off, and all of a sudden I'm bestest friend (apparently) with big beefy Nord dude...I definitely want to get somewhere where there are witnesses around, and I can scarf a mead or many in peace.

Spin round and round when I get outside of tutorial cave, then walk in the direction I'm facing when I stop.

Lately, when I start a new game, I avoid Whiterun for a while. I didn't get in there till I was 21 I think on my assassin-thief, and the only reason to get there was Thieves guild quest.
I usually get to Riverwood, and than I just wander off in random direction, depends what type of char I play.

I tried not going to Riverwood. Ended up going there anyway. It's a really good starting place. If you play your cards right, you can leave there with a couple thousand gold before heading off to Whiterun.

I never stay in riverwood, i always just cut trough the river,then over the mountain and arrive at white run stables.And depending on what character i make i just go to a city and start the quests there.When you get out of helgen you have around 700 gold,which is more then enough to get you going.