Alright, now we step into a realm that may not be as much fun as dealing with the occasional vandal, but instead one who continues to vandalize, make attacks against you, and whose only intent is to harm wikipedia.
You occasionally get the repeat vandal. The vandal who is here, not because he is bored and has nothing better to do, but because he has a singular purpose of wreaking as much havoc as he can before he gets blocked. These vandals go in and remove entire sections of text, or replace entire pages with gibberish repeatedly. Even after you've given them a warning, they ignore it and continue. It is for these vandals we have multiple levels of warnings. Most of the warning templates come in four levels: (following from WP:UTM)
In general, you will escalate up those levels from 1 to 4 as the vandalism continues. If it's nothing clearly malicious (see below), you should always assume that it was a careless mistake (in short, assume good faith, one of Wikipedia's foundation principles), and just let them know that you fixed it. As it continues, it becomes more and more obvious that they intend to cause trouble, so the warnings get more and more stern. Occasionally, you'll get the vandal, who despite all logical reasoning, continues to vandalize after that final warning. When this happens, we have no choice left but to block them. Since we're not administrators, we lack this ability, so we must report them to those with that power at Administrator intervention against vandalism. That page provides complete instructions on how to file a proper report. If you ever re-install Twinkle, you can report a user to this page by clicking the "arv" tab at the top of any of their user pages. Usually, an administrator will take action within minutes, but until that happens, you need to continue watching the vandal's contributions and reverting any further vandalism. The Three-Revert Rule does not apply when dealing with obvious vandals.
Then you get the belligerent vandal. This is very similar to the last kind, although they actually take the time to read the warnings (or are able to) and take offense. They go by the logic that anyone can edit Wikipedia, so who are you to tell them that they can't edit in this particular way? To make this rather annoying point, they will leave an offensive message on your talk page, or more often simply add some sort of vandalism to your main user page, which you generally won't notice for several more minutes, or days, if someone else reverts it first.
When this happens, particularly that last version (although those are exceedingly rare), you just have to take it in stride, and remember that you are far more intelligent than them because you actually stop to read information instead of blanking it away, and thus the human race still has some hope for salvation. Just revert it, and slap them a ((uw-npa)) warning of whatever severity you deem necessary. The last version got a ((uw-npa4im)) warning, an "only warning" for the most severe offenses, and I still reported him straight off anyway.
The final version is the malicious vandal. These are hardest to notice, because their edits aren't immediately recognizable. They will seem to be improving the article at first glance, when really they're replacing true information with false, often libelous parodies. Others replace valid links with shock sites, or add hidden comments with offensive information. This last version doesn't actually appear in the article, but is there waiting when someone comes to edit it. A similar type of vandal, the "on wheels" vandal, is here for the sole purpose of destroying the encyclopedia. The namesake, Willy on Wheels, replaced dozens of pages with the text "((BASEPAGENAME)) has been vandalized by User:Willy on Wheels!" The BASEPAGENAME variable is a magic word that displays the name of the page. After his blocking, Willy continued to create hundreds of sockpuppets for the same purpose. This sort of vandal is clearly here to vandalize, as such actions are not accidental. With them, you can safely assume bad faith right from the start and slam them with a more severe warning. No, you don't have to escalate. The "4im" level is designed for initial cases of severe vandalism, an only warning to cease and desist.
Keep an eye out for all of these vandals, and keep that information in mind when stopping them. There is a full customized range of warning templates to be found at WP:UTM - use the most specific one possible, so that the vandal, if he did make a simple mistake, has the links at hand to learn from his mistake and improve. Any questions, please put them on the adoption talk page.
Perform some recent changes patrols, locate and revert vandalism, then give the vandal the appropriate level of warning. If you are unsure the edit you are reverting is vandalism, then do not revert it! If you come across a vandal who needs attention from myself or another user/admin, leave me a message, or leave a note on WP:ANI with a clear explanation of the situation. Remember if the user vandalizes after a final warn, make sure to report them at WP:AIV. Good luck, let me know when you have completed....oh lets say more than 100 good reverts.