Do I just use random ingredients until I make a potion of minor healing or whatever? I walk around the fields and harvest a ton of stuff, but when I'm at the alchemy table it seems hit or miss whether or not I actually make potions. Alot of times it's potion failed.

Is there a list of the recipes on how to make the different potions?

Eat ingredients to find out the first effect - you'll soon be able to make plenty of potions. And sometimes when you make one of those potions, you will discover other effects of the ingredients you combine.

Edit to add: you can also buy recipes from alchemy vendors, they are quite handy.

http://www.giantbomb.com/guides/skyrim-alchemy-guide/2310/

Well, first, some NPCs to sell recipes telling effective combinations for some potions.

Second, yes, you can just experiment randomly.

I believe just eating the ingredients also works to a certain extent.

Once you do determine what the effects an ingredient have are, the game does organize them by effect.

http://elderscrolls.wikia.com/wiki/Alchemy_%28Skyrim%29#Alchemy_Effects

The alchemy shop in Whiterun is it? The first town? Under books she sells some recipies for alchemy. Hope this helps out out. But otherwise just eat one of each to find out an effect and just mix some around and hope for the best

Nice. Although I don't see the potions of magicka or health. You know like the ones you find in dungeons, etc. We can make those too I assume?

They are called Restore Magicka and Restore Health. They are on the list.

Oops. Thanks =)