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.
Smart casting doesn't mean "It'll cast stuff for you". Smart casting means "When you hit the hotkey, it immediately casts the spell as if you were targetting where your mouse cursor is hovering".
When you hover over a unit, the colored border around it will thicken. This will denote exactly which unit you will smart-cast against when using a targetted skill (like Rammus' Puncturing Taunt), provided it's a valid target.
When using an area-of-effect or skillshot skill, then a smart-cast will be targetted as if you had clicked whereever the mouse is currently situation.
It saves you not merely time in setting up the launch, but also gives you much more flexibility in the actual gameplay. Using normal targetting practices, you have to either select your target or cancel the skill, you cannot input any other commands. With smart-casting, you can focus on the setup for your skill, and then adapt your movements to any updates in the situation without needing to cancel out. And then once you get in position, it's just one button click to fire. Since it can be thrown out from any action as fast as you can drag your mouse over, it can be a lot faster and catch your opponent off-guard, especially for otherwise predictable lead-in skills.
Its utility in being smartly used is entirely dependent on how effective you can utilize aiming without the guide ranges provided for skills. For targetted effects, it rather depends on your ability to target by hovering on a unit rather than clicking on a unit (so in clusters, it is important for knowing the timing so that you don't cast on the wrong foe). It offers speed, but unless your accuracy is up to par, it may be more effective to stick with normal casting for skills until you get the hang of it.
Best Answer
The question depends on what you want your role to be... or, if you're getting stomped in lane, what role you can be ;-)
Death obviously grants you a ton more damage by giving you the DoT (Damage over Time) effect. In fact (from League Of Legends wikia):
That's... a big deal. Especially for an AoE ability.
However, it does leave Gravity Field a little weak because of how vulnerable you have to make yourself to cast it into a teamfight. Moreover, Augment Gravity doesn't just make Gravity Field stronger:
Now that we know the benefits... Which one is better?
If your team is relying on you to dish out a ton of magic damage, for instance if you have a tankier team or an AD-heavy team (forcing the opponents to build armor to cope), then you almost have to get Augment Death. Viktor's entire damage kit was balanced around that idea. If you don't, you'll be hamstringing your role.
However, if your team lacks a lot of CC; has an initiate that brings the enemy team into a tight formation (Shen taunts, Orianna ults (because you have two AP? I don't know); your team has a lot of gap-closers (Irelia, Xin Zhao, Jarvan, etc.); or if you're just behind and need to be helpful in a different way, you want to get Augment Gravity.
TL;DR: Augment Death lets you crank out the damage, but your team utility will be low. If your team needs the additional utility, or you're too far behind to do meaningful damage, Augment Gravity.