I love starting characters, so I have a few methods.
With the Helgen start, I most often leave Hadlof at the cave and run back through Helgen, north past Orphan Rock, the Stormcloak camp (which is not here yet if you go this route, but the quartermaster already is, standing all alone), Haemar's Shame (duck in if you want to play a vamp or just loot some early magic armor), and the Alchemist Hut to get to Ivarstead. If I'm feeling lazy I'll pick a lock in Ivarstead and take the 5 septim Justice Express ride to Riften, where I have a carriage, TG stuff, kick off DB if I want, start the Mara quest for a quick 15% MR, and hunt skooma dealers so's I can buy Honeyside (Honeyside is most often my first house, not Breezehome).
I have a mod called Alternative Life, which gives you a huge range of starting options (including the regular vanilla start). If I'm not planning to start the main quest right away (or at all), I choose "Attacked and left for dead." This option drops your character at one of five or so isolated spots around Skyrim with nothing but some clothes. I keep my fingers crossed for the hot springs region, because then I can grab the Atronach Stone, Akatosh shrine, word wall with boss chest, huge amounts of jazbay and dragon's tongue, and several chests and both fixed and random encounters. From there it's easy reach to Darkwater Crossing (smelter), Kyne's Grove (smelter, mine, inn), Fort Amol (for bound bow) then Windhelm and a carriage.
Back to Helgen start: If I head to Riverwood I always stop and take out that little bandit camp. This gets you a selection of light and heavy armor and a treasure map to an extra chest in Riverwood. I'll stop at Embershard and kill the bandit out front, but usually will be too loaded down at this point to go inside yet. Run to Riverwood, talk to Sven, rat him out to his ladyfriend, sell all the junk, get Faendal as a partner then it's off to Bleak Falls Barrow. I don't bother getting any quests yet usually. They're just extra dialogue you don't need to complete it. From BFB exit, mine three corundum veins nearby then back to Riverwood, give Lucan his claw, talk to the town leader whose nephew you chose, then it's over the hills and far away to the big city. Give stone to Farengar, buy Breezehome, then (usually) leave Irileth standing by the guard tower waiting for me for the rest of time, because I forgot all about it and the only reason I went and got the stone was to buy Breezehome.
Once I get a carriage, I'll travel to Dawnstar for a run to Lord Stone or Solitude for a run to Steed Stone (if I didn't pick up Atronach), then usually travel to Markarth because it's my favorite city with great light armor to be had for a quick quest, or great heavy armor from materials literally lying about the place.
Occasionally I'll go the other way and get to Falkreath first, but this is actually my least favorite city. No smelter, no carriage, the Jarl's a dick, the blacksmith won't stop jabbering about some mutt and won't sell to you unless you agree to go fetch it, and pretty slim pickings for quests unless you are a follower of Hircine.