you can do this at the start of the game, go with the imperial guy and have him free you and when you get to the first town just wait in Alvor's house till 8 am and only the soldier should be in there, then sneak and power attack him to your hearts content... Also if you are leveling a thief or Warrior type make sure that you touch the stone that you most want... I'd suggest warrior since sneak attacking this npc will level your sneak quickly with out the 20% increase.
You can do it earlier than that.
Remember the part of the Helgen escape where your rescuer wants you to sneak past a bear?
Yeah, don't do that. Instead, sneak behind him and start sneak-attacking. Sidenote: power-attacks don't level your sneak any faster than regular attacks, so just regular attack if you don't want the guy to stagger all over the place.
I was level 10 with 70 sneak skill before I ever left the cave.
You can also level weapon skills this way, but that takes much longer. Faster weapon skillups to ignore the sneak-attacking and just beat on the guy as fast as you can, since every hit gives you a little weapon skill. I think I had about 32 one-handed skill when I left the cave starting at about 20.
To the people having trouble with wispmothers and ice wraiths as fire mages: shame on you. Spam Fireball right under them, like you'd use grenades/rockets in a shooter, the splash damage is just as good as a direct hit. If you're up to expert-tier casting, use wall of fire + flame cloak instead.
My mage hit 82 destruction not long ago. As long as you keep using the highest-level spells you can reasonably use it'll keep going up at a reasonable pace. If you just use the Novice Flames/Sparks/Frostbite spells it'll be very very slow, so start using Apprentice+ spells as soon as you can.
Also, ways to level non-destruction casting skills quickly:
Restoration - use Healing and Equilibrium (found in the Labyrinthian, not sure where else you can get it) at the same time. Equilibrium drains your health to give you mana, Healing uses your mana to heal you. Just make sure you don't accidentally suicide as EQ drains faster than you'll be healing. Once you get to Adept and can cast Close Wound you'll be able to out-heal Equilibrium by spamming it and skill up even faster.
Conjuration - easiest one to max out. Get Soul Trap, cast it infinitely on any corpse, gain as much skill as you want.
Those are the only two I recall. Might add more caster advice later if I come up with anything useful enough to share.