The Witch is an extremely versatile role whose power is determined by the combination of roles in the setup. Its potential ranges from relatively lackluster to extremely powerful. The role has the ability to control a target at night, selecting who they use their night ability on. The Witch wins if the Town loses and the Witch is still alive.
Generally put, play similarly to a Doctor by avoiding being lynched while also avoiding attracting any attention during the night. Two distinct disadvantages from a Survivor are that 1) You cannot win with the Town and 2) You have no bulletproof vests.
Later on, if you're sure who the Mafia, Cult, or Neutral Killing roles are and if you think you can gain an edge by making yourself known, then it's better that you claim your role correctly before anyone else does as there is no real way to prove that you're a Witch.
Doctor - Seizing control of a Doctor is extremely beneficial to a Witch as it renders the Witch immune to death at night as well as leaving the Town vulnerable to having their important roles killed. Extreme caution should be advised in ensuring that the target is indeed a Doctor and not a role that will be harmful towards the Witch.
Sheriff or Investigator - Controlling a Sheriff or Investigator essentially makes these roles entirely useless as they can simply be forced to target themselves over and over again. This stops the Town from gaining any real leads and will almost certainly lead to their demise.
Controlling an Investigator is less beneficial because they will realize that they are being controlled because they get the same message every night. Try making them target random confirmed Townies to make the fact that they are being controlled less obvious.
Sheriffs are better to control because they will get the "NS" message every time they investigate someone. The Sheriff will only realize that he is being controlled after he investigates everyone and they are all "NS"!
Take care when controlling killing roles, because you would not want to have a Mafia kill another member of the Mafia, lessening their chances of them (and you) winning.
You can control Mass Murderers in a similar fashion. Just make sure you do not force them to target themselves as they will camp their own house. Because you have primarily targeted them, the Mass Murderer you controlled will kill you.
You could also coordinate that Mass Murderer. Tell him to keep targeting himself, and every night make sure you send your target to visit the Mass Murderer, resulting in your target dying that night.
Framers, Janitors or Arsonists - Can be controlled, but generally difficult to use effectively if at all. Framers and Janitors will be working against the town regardless and Arsonists are next to impossible to detect as a Witch until they've ignited their targets.
Siding with the Mafia - Whilst the Witch is essentially a Neutral role siding with the Mafia can be extremely beneficial as both the Witch and the Mafia have a common enemy. It is often a powerful strategic move for the Witch to let itself be known to the Mafia so that it can help them eliminate the town.
This becomes much easier if Private Messages (PMs) are enabled. Also, faking being blackmailed can help promise that the Mafia will not kill you at night, and may even lead them to suspect you're an evil role who is siding with Mafia. To fake being blackmailed, simply spam the vote button. In most scenarios, people will assume you are blackmailed. Take care with this plan, however, because this may force a Investigator to check you and discover your role.
When a Veteran is in play, you can also control [i]anyone[/i] and make them visit a revealed Veteran. It can be a Spy or a Citizen for all it matters, witching them and making them target the Veteran will indeed force them to visit him/her, and resulting in them being killed by the Veteran. A useful strategy when you need to kill off a Town role, without forcing a Mafia or Serial Killer from attacking them. This requires great caution however; this will reveal a Witch is in play, and can serve against you if you're a silent Witch, though only if it's a role that cannot normally visit someone alive at night. This includes Mason, Citizen, Spy, Mayor, Executioner, and Survivor.
Controlling a Coroner will actually be more harmful toward yourself than to the Coroner. Although it wastes one of the Coroner's nights, this is because Coroners can only successfully autopsy dead bodies. Because you can only force a Coroner to target a living player, the Coroner will be notified that the person he tried to autopsy is alive, therefore revealing that there's a Witch in the game (even if targets don't know that they are manipulated).
A Witch can not prevent non-targeted actions such as using bullet-proof vests or igniting gasoline (Survivor, Citizen, Arsonist)
A Witch cannot force their victim to target dead people. It will make Coroners target a living person, but mind-control has no effect on Amnesiacs
Even though Vigilantes themselves cannot willingly kill on Night-1, Witches can still force them to kill other players.
The Witch can not force roles that are immune at night (such as Godfather or Serial Killer) to commit suicide.
The Witch can force a role that does not normally have the ability to 'visit' (e.g. Citizen) to visit someone, showing up in Lookout/Detective reports.
Witches win with Survivors, keeping them alive can benefit her, though in most situations she will have to force the Town to lynch the Survivor instead of her late-game if Mafia has lost majority or is currently dead. Witches also win the game with Arsonists and Serial Killers.
If a Witch controls a second Witch to target a third Witch. The first Witch controls the second Witch to target the third Witch. Witch target-actions are the people they control, not the victim's new target. So, the second Witch will control the third Witch and make him target X, while the third Witch will control X and make him target Y. X and Y are the people the second and third Witches originally ordered their victims to target. 
Amnesiacs are immune to manipulation. However, neither the Witch nor the Amnesiac will gain any feedback.