My path is pretty much like sulu's, though I use Skyrim Unbound, and depending where a given girl is going to have a house and a business (which all of mine do), I'll generally either start in that area, or go there shortly after running the "early route" of bandit extermination similar to what sulu listed. I don't buy training - I help Adrianne or one of the other smiths (depending again on girl and area), then craft my way up a bit in smithing and enchanting. I get plenty of "training" in one-hand just killing bandits - by the time I get through that route, one-hand is usually around 35, and by the time I get smithing up a bit, I can Superior my katanas, which lasts for a good long while.
I pick up all the books you can get training from - there's the one in the bandit camp down the back way from Helgen, that's Light Armor; the one at the first bandit camp the other way from Helgen is One-Hand. There's a few out in the plains beyond Fort Greymoor - one for conjuration I think, one for speech, one for.... hmm, not sure - illusion maybe? on a body on an altar guarded by skeletons, one in a chest under a puzzle-opened cover near standing stones - that one might be enchanting, again not sure (haven't done those on the current girl yet). After I get the books around, I mostly just quest and get skills as I kill during the quests. I don't use magic much so the books on magic just add a bit to make the first few levels or so a bit faster. If I get to level 15 or so and Light Armor's a bit low, I'll either hunt down other books for it, or just buy some training at that point. It's about the only thing I ever buy though - speech does itself as you sell stuff you don't need, as well as doing some of the speechified quests. I don't do alchemy, don't like it, the only points I ever have in it are from books I find - there's one in Dead Crone Rock, and one in Forsaken Cave for instance. And enchanting I level up just doing my own gear....