Alchemy - Get the equipment (mortar & pestle, retort, calcinator, and um the other one) - don't worry about what they do, the higher quality the better)
Now select one of them in your inventory and you get the potion making screen. Add an ingredient. You will see as you scroll through your ingredients that they have different properties. What you want to do is choose two ingredients that have the same property (eg. restore health, fortify magicka) - If you do this right, then the right hand pane will say what sort of potion will result. Click create potion and you get one (and your alchemy level goes up)
As you get better at alchemy, your potions get stronger and you are allowed to see two, three, then four of the properties of the ingredients. This gives you more choice of ingredients, and at highest levels, some really cool potions that have multiple effects.
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ASoul gems have two purposes - recharging enchanted items and creating enchated items. To fill an empty soul gem, cast a soul trap spell on a monster (not a human) and then kill it before the spell wears off. I usually leave Soul Trap until just before they're dead. When they die, their soul is caught in the appropriately sized soul gem. Always carry a selection of sizes, so you don't accidentally fill eg. a Grand gem with a Petty soul. Human souls can only be trapped in Black soul gems (which can sometimes be found wherever you find necromancers)
To recharge an enchanted item, click a full soul gem - you get a list of your enchanted items and can click on any one to charge it. I forget how much charge each one does, but I do know Grand=1600 points.
To create an enchanted item, you must gain entrance to the Arcane University (join the mages guild and do the reccomendations) - there, at the praxographical center, use the altars to bind a soul gem to an item to create magic items. You can do anything you already know the magic for and the power is limited by your mysticism skill and your money (it can get pricey)
EDIT : And yeah, Read the manual - it's all in there