Map Awareness
Riot has garnered some pretty effective videos with their Video Tutorial contest (Links posted below). As a beginner, you want to primarily focus on making sure you don't get ambushed by the other team (called a "gank"). While some of this involves Map Awareness, you also need to learn to not overextend, as well as keeping an eye on the enemies you're fighting in your lane. If you cannot see one or both of the enemies in your lane, let your teammates know!
Last Hitting
Killing minions and enemy players nets you gold to buy items. However, if you don't land the killing blow on an enemy minion, you will get ZERO gold, even if you've done most of the damage to it.
The faster you kill enemy minions, the faster the minions reach the enemy tower, which will make quick work of all non-cannon minions. It's better to keep the minion fights as close to the middle as you can; you only need to finish off the minion to get gold from it -- see if you can limit yourself to only attacking minions when you will kill them!
Tower Aggro
The enemy towers are dangerous. They have lots of health, armor, and a powerful attack that can quickly decimate low-level heroes. It's important, then, to know how the turret picks its target.
Enemy towers will always target allied minions first, switching to champions only if there are no longer any allied minions around. The exception to this is if you deal damage to an enemy champion, in which case the tower will immediately switch to the damaging champion, shooting at them until they die or move out of range.
Let me repeat that: If you deal damage to an enemy champion within range of the tower, it will immediately start shooting at you: You do not want this. Be very careful when attacking the enemy at their own tower.
Recommended Viewing
Four of these videos are a result of the Valoran Video Contest held by Riot Games, and are quick, 1-minute videos touching on just the basics (which seems exactly what you're looking for). The fifth video is made by Shurelia, an employee of Riot Games, and goes much more in-depth into what "Zoning" is, and how to take advantage of it.
Roaming has become a popular addition to the metagame strategy, wherein one player does not lane nor jungle, but rather simply moves through the jungle/river to initiate ganks on lanes from level 1 and onward, until the teamfight portion of the game begins.
A roamer essentially guarantees that you're either running a solo lane, or two solo lanes if you have a roamer and a jungler. I've also seen a double roamer game played, though this tends to be less effective as enemies who are wise start playing defensively and reduce the effectiveness of your roamers, which put them behind in both CS and levels.
The qualities of a roamer can vary, but the general requirements are that they are mobile, can do burst damage, and have a strong stun or position altering (push, throw, etc.) ability. Taric, Evelynn and Alistar are all very popular choices for roaming. I've also seen Sion and Blitz play roaming effectively, but they were both out of a lack of a jungle spot on the team and normally suit better in a lane.
This strategy is similar to the "double jungle" mindset in that you get three solo lanes, with the exception that there is extra pressure on getting early kills to succeed. A definite risk vs. reward tossup, as successful roamers will quickly catapult your laners ahead in level / CS as well as demoralize your opponents, but unsuccessful roamers will fall just as far behind due to the lack of their presence in the laning phase.
Best Answer
First and foremost, practice your last hitting. That is the single most important thing in any lane role in League of Legends.
Pick a champion that seems right for you. Right now, the popular ones are Kassadin, Ahri, Sion, kennen, Ryze, Karthus, Morgana, etc. After you pick your champion and are ready to practice, learn their auto attack animation and get comfortable with it. You can push A then click on a minion to force your character to attack a certain minion the millisecond your champion can autoattack.
As for masteries, I would run 21/9/0 grabbing all the AP/damage you can in the Offensive tree, as well as magic resist in Defense with the extra health. Runes should be Magic Penetration reds/marks, Ability Power per level yellows/seals, and Ability Power per level blues/glyphs with flat AP quints.
Your build usually starts with boots and 3 health potions, as the mobility to farm, evade ganks/harassment, and harass back is super duper important, as well as it enhances your ability to go gank other lanes. And as a tip, always build to win your lane. If you start taking too much damage and keep getting CC'ed by your opponent (or their jungle) and there is nothing you can do, grab merc treads for boots.
The magic resist and CC reduction is well worth giving up extra damage. but I'm going to go my usual build from farming and a normal not incredibly fed game. I usually go 2 Doran's Rings into magic pen boots, into a Deathcap, Void Staff, and then defensive items or more AP damage to either nuke harder or to help keep yourself alive.
If you happen to get super fed, go 3 Doran's Rings, into magic pen boots, Deathcap, Void Staff, into more AP. But these builds are just the normal AP route. Different champs call for different builds.
To put a short version on this:
Those are the things that go through my mind when I solo mid as an AP. I hope its not too confusing and gives good pointers.