If you're early on, save those filled Common gems. Lesser and Petty gems are easy enough to fill, and are great for grinding up Conjuration: Soul Trap is an Apprentice-level spell, and unless you have the Conjuration Apprentice perk, Soul Trap will hit your magicka bar up pretty hard. Petty and Lesser are great refill fodder, and come from baddies who usually aren't much challenge. At low-level, Common might be a good "My First Enchantment" gem (to make an enchantment, not just refill a weapon enchantment). After a while, you'll graduate to Greater souls for enchantment creation, and then the top-grade Grand gems. Often times, it may be less hassle to simply buy such gems pre-filled from mages of the Jarls' courts (who double as magic goodies vendors) or College of Winterhold instructors, or (less commonly) from general goods vendors (which includes the Khajiit caravans).
Black gems? There's some caveats on their use. Basically, they're inferior to Grand Gems in every way except one: they're easier to maximally fill. Grand Souls usually are in big beasts that will stomp you flat. Black gems? Any pissant bandit will do, and you'll get a Grand-size soul. Beyond that, they're rare to find, aren't sold by reputable vendors, weigh twice as much as a Grand gem, and are technically contraband... and for a very good reason: nasty horrible things happen to those souls. If you have Dawnguard, you'll find out the details in the course of the Dawnguard main quest.
And on the subject of making your own enchantments, you'll need to find an enchantment altar and have an enchanted item you just found. You'll have to "break" that item to learn the enchantment, but after that you can apply that effect as many times as you have gems. Higher enchantment skill and perk advancement will yield higher maximum effect magnitude and higher charge capacity. If you want to smith up that weapon, you'll need to do that first, as enchanted weapons require special care via a mid-high level Smithing perk to do after enchanting.
If you're early on and can only get 5-magnitude elemental destruction enchantment items out of loot, you definitely will be interested in a DIY enchant: the lowest maximum enchantment magnitude for novices at the start is 10. You can lower that from the max to increase the number of discharges it can hold (as would using a bigger gem). The nearest enchantment table along the early path of the Main Quest is http://www.uesp.net/wiki/Skyrim:Farengar_Secret-Fire's at the beginning... or if you're okay with breaking and entering cellars and killing old ladies who are witches, there's a little hut west of Riverwood with one too.
A beneficial side effect of grinding up Conjuration with Soul Trap is the Summon Atronach spells. Raise Undead spells take too long for the corpse to get up on its feet and fighting, and requires the presence of dead bodies. Atros? Flick your wrist at a spot near you, and you get an elemental daedra (temporarily) on your side.