Increasing Stamina lets you sprint for longer amounts of time and it increases the amount of power attacks you can do before your stamina is depleted. I believe you have to find an alchemy table to mix potions. You can usually find them in encounters with mages, or any place to do with magic. The College of Winterhold is one of these places, for example.
Disclaimer I have not played the game.
However after reading both yours and GOA's posts I don't really see that much of a problem. He's having an easy time with something , ok it's a dragon and it should feel like a fight because his character has the right sort of build. Mages kick his ass because he has the wrong sort of build for fighting mages.
On the other hand your character is not built for fighting dragons, but probably has no problem killing mages ?
Without knowing why the Dragons are so easy for GOA it's hard to pin down. But two characters with different builds finding different things easy/hard. That seems like good RPG design to me.
It dosen't have anything to do with builds. Its because of the broken level scaling in Skyrim. Dragons were made easier so the player has a better chance of progressing in the main story and unlocking shouts. However the same can't be said for the rest of the hostile NPCs. As you progress and become a higher levelled character, the NPCs become stronger as well. So being a level 28 Dragonborn dosen't matter to a Frost Troll, it's still going to kick your ass because its at the same level as you.