Sadly, there's no sure-fire way of resolving this situation.
You won't be able to rely on minions to take out their towers and nexus, especially if their uber status gets to the point where they can start defending their respawning inhibitors. If you get really lucky and get their nexus towers down, you could just farm minions hard enough for yours to take the nexus down a bit then try to dive the nexus. I'd consider this a Hail Mary more than anything else, though.
If you know you're going up against a team that is going to steamroll you once they get maxxed out (2+ carries), the main goal is to take advantage of the fact that they will be weaker in the early game. If you zone the carries and/or kill them enough to keep them from getting geared and leveled, your team will get powerful enough to take their nexus before they can do anything about it.
Assuming you find yourself in that unfortunate position anyway and forcing their team to split up isn't working, you absolutely have to focus the primary damage dealers down first. If they stick together and you're not killing their carries first, you'll take way more damage than you will deal out over the course of the fight and you're sure to lose.
Yes the champions can be categorized by types and roles, as the types are less i shall start with them. de facto there are two types of champs.
melee
ranged
these champs have each an affinity to one of the following, even though each will be set with any champ, and depending on how the "affinity" is mixed (sorry not native english speaker) they are given different roles
abilities
auto-atacks
damage-soak
usually ranged champs are not affine to absorbing damage.
melee champions who specialize in damage-soak and have lots of Crowd Control (disabling the enemy) are usually called Tank. Their job is to protect the Carry a (usually ranged) champ either specializing in abilities (AP-Carry/Caster) or auto-atacks (AD-Carry)
champs who specialize in dealing damage to the enemy Carry are called Bruiser or Assassin. these champs usually have some skill to close up onto the Carry and deal quite some damage to kill him quickly.
The big difference is: Bruisers are more like Tanks and Assassins are more like Carries
There are also champs who are designed to provide lots of Utility to the team. they usually have Buffs and Heals at their hands and are called Supporters.
Likewise to DoTA there is a Jungle in LoL. Champions who can (not must) sustain well and have a high Ganking-Potential (meaning they can Crowd-Control and / or close gaps well) are called Junglers. a Jungler usually is either a Tank, Assassin or Bruiser
Final Conclusion:
There are 4 major types of champions:
1. Carry
2. Tank
3. Assassin/Bruiser
4. Support
Best Answer
Sort of a tangent, but if you plan to successfully climb the solo queue ladder, you will have to play AD carries. Not to mention that understanding the aspects of all lanes is essential regardless of which role you fill.
Animation cancelling is more prevalent in other games such as StarCraft:BW and DOTA, but it is still useful in LoL. Basically it will reduce predictability in lane when you're last hitting as a ranged champion (generally mid and bot) as you're standing still for a fraction of the original animation time.
Personally, I come from a SC:BW background and animation cancelling is second nature so I didn't have to spend time practicing it, however all it takes is realizing how long the attack animation for each champion lasts. Once you attempt to cancel it and fail a few times (by cancelling the entire AA and consequently its damage), you'll pick it up quickly.