There's a path to find the Greybeards that requires zero climbing. You have to go to Ivarstead, and there's the '7,000' steps of peregrination to reach the temple of the greybeards. There's a guy that gives you a quest to deliver some food to the Greybeards, who leave off of donation (Neckbeards?), and there's also a side-quest that's semi-hidden to read 10 stone tablets on the way to the Greybeards, which gives you a bonus for a good while that IIRC, makes it so that animals don't attack you. I might be wrong on that one though, it's been a while.
Things i'd make a priority on a new playthrough:
Fuck right off to Riften as soon as possible. Easiest way is fast track to Whiterun after the mandatory intro, then take the charriot to Riften. On the Riften docks, there's an Argonian by the name of From-deepest-fathoms who is a little bit on the crazy side. She'll give you a quest to go to a dwarven ruin, pun-intended, the name of the quest is "Unfathomable Depths", clever Bethesda is clever.
Dwarven Ruins are an AMAZING source of cash/xp and gear. Dwarven gear is one of the best gears available in the early game, and levelling up Smithing to smelt all those dwarven ingots will give you plenty of cash to spare in the early game. You'll also gain a VERY useful bonus by completing the quest - making Dwarven Armor 25% more effective (effectively putting it on par with Ebony armor - prrrreeettty good) - and also making Smithing go 15% faster.
Which is why i'd make it my top priority to do this quest.
Also in Riften, there's the Thieves Guild. Which is pretty easy to speedrun through. The quests aren't ACTUALLY that hard, and you get some VERY neat perks, like fences who will buy your stolen goods - no questions asked - a infinite source of lockpicks, as well as a free base to call your own. The light armor/gear they give you is ALSO pretty good.
ALSO in Riften, there's Mercurio, arguably one of the best companions in the game, which you can hire for about 500g. And another companion that's a meatshield, Mjoll the Lionness, which is also a quest-giver. Her quest is pretty easy, and once you complete it, she's your follower for free. BONUS: The quest involves going on another dwarven ruin, so MORE dwarven loot to sell/smelt into ingots then smith and then sell.
By that point, i'd then abandon Riften for a while. No more super amazing quests or gold to be found, unless you want to go into the Black-Briar Mead stuff, but that can wait.
Markarth would be my next stop. Through carriage, you can get there pretty easy. There's SEVERAL amazing quests in Markarth. And, as a bonus, several GOLD AND SILVER mines nearby. SCORE!
Don't EVER underestimate mining in Skyrim. Mining is VERY useful, and can net you a very nice profit. Gold and Silver mines are one of the most profitable and most important in the early game.
Why do you ask? Well, the game decides your Smithing XP by the value of the item you're smithing. Jewelry is expensinve, dawg. Which means Jewelry gives some of the best XP you can get through smithing. And it doesn't require many perks at all to get into.
But don't go smelting all your silver/gold ore just yet. Transmute Mineral Ore is an actual spell in Skryim and can be found in Halted Stream Camp - close-ish to Whiterun. It's also a pretty good Iron Ore mine.
Did i mention you can transmute Iron to Silver and then to Gold?
You see where i'm getting at with this.
Contrary to what the game might lead you to believe, the fastest and best way to raise your damage in the game is by raising your smithing to godlike levels. Smithing can be raised very very fast, and it gives you real damage increases on your weapons right off the bat, as well as increasing your defenses the same way. And you make a quick buck by converting near worthless Iron Ore into pretty expensive jewelry, which you then use to buy more Iron Ore...
In a few short hours you should be able to hit Smithing level 100 no problem. You then get Dragonbone weapons and armor and are basically pretty much invincible to anything until the enemies level catches up to what you can muster in way of power.
This is what i'd do if i was in a new playthrough of Skyrim. This will make you powerful enough to be able to explore everything without much concern.
Another quick tip i'd give you is to use the caravan outside major cities to travel between all of them so you can get them all on your map for fast-travel purposes. You then never need the Caravan again. Also in Riften there's a quest to steal a horse called Frost which is a pretty good horse, and IIRC you can use Frost forever if you never hand in the quest. Do make him essential, though. He's not by default, and as such, can die.
The Dark Brotherhood though, can get you Shadowmere, which is actually immortal and looks super-badass.