I'm the owner of the wiki  http://teflpedia.com and people sometimes ask me about setting a wiki up - so here are some thoughts and ideas about the process. (Some of these notes were previously posted on teflpedia)

In truth, if you are thinking about starting a wiki you will probably already be aware of some of the following, but it is written for those who are totally green.

In general, you have a number of options with regard to the software and the hosting.  Typically the cheaper and easier things are the less control you have over the technical side.

Wiki farm

The simplest way to create a wiki is to use a "wiki farm" like http://www.wikia.com, http://bluwiki.com/go/Main_Page ''bluwiki'' or http://www.wikidot.com/ .  In order to create a wiki in such places you may first have to present a proposal, or you may simply be able to start one up.

In general they will have some way to monetize the process.  In the case of ''wikia'' they will put advertisements of their choice on your wiki and you will have to pay to get rid of them.

Such sites vary, but you will have less technical control than you would have over a full Mediawkiki installation, on the other hand you should also have fewer technical problems.

Before using one of these sites you should check up on the possibilities of subsequently moving your wiki to another site in case the need arises in the future.  For instance, if your wiki becomes immensity popular and you wish more technical control of it for some reason.

It should be borne in mind that the wiki farm you choose may cease operating at short notice, or even no notice, as happened with http://www.wikiindex.org/ScribbleWiki ScribbleWiki and http://www.wikiindex.org/ElWiki ElWiki.

Simple wiki

If you want to be more technical you can set up a different type of the Mediawiki software.  Some of these are simpler to operate than a full mediawiki installation.

You can find a list of the various wiki programmes at this http://en.wikipedia.org/wiki/List_of_wiki_software  Wikipedia link.  Of course the majority of your potential users will be familiar with the Mediawiki installation used at Wikipedia.

Mediawiki

Finally you can take the bull by the horns, buy some hosting, and set up your own mediawiki installation.  My guess is that this is what many people will want to do as that will give you the same “look and feel” as Wikipedia.   But it’s a trade off of low cost and simplicity against higher cost and more complexity.

Another advantage of a Mediawiki installation is, that for people used to Wikipedia, it will quickly begin to look like a "real" wiki.

Other considerations

Before starting up there are various other issues you will need to consider.

Monetization

Another point to consider is the future.  Let’s imagine that your wiki becomes amazingly big and successful and you think that you could be able to make money from donations or  advertisements – or even sell the project.  If you’re locked into a wikifarm then that will not be possible.  You would also want to consider the implications of running a non-Mediawiki installation.

Copyright

You will also want to decide on the copyright status of the wiki.  Currently many wikis are under  Creative Commons By-Attribution Share-Alike license as that means they can share content with Wikipedia. But that means that the contributions to the wiki don't actually belong to you - whether this will cause you a problem in the future depends on your ideas for the future of you wiki.  If you are prepared for people to use your content without any attribution whatsoever then you can make it ''Public Domain''. It may be easier to start off Public Domain and go to a more restrictive licence later than go in the other direction.

Namespaces and categories

While you can set these up after you have your installation up and running it is a good idea to spend some time thinking about it beforehand.

Purpose and limitations

You should think about this and have it clear in your mind before you start. You can always change things later but you should begin with a clear written goal.

Cost and Teflpedia's hosting company

So let’s imagine that you want to set up a mediawiki.  You need a host.  Teeflpedia uses http://www.eukhost.com/ UKHOST's copper plan which is a surprisingly reasonable  £29.99 a year. That was fine for Teflpedia's needs but when Teflpedia hosted another wiki for a while it wasn’t enough.  You’ll note that the next level up is 65 pounds a year.

Teflpedia's owner has always found their "live chat" people to be very helpful and there seem to be some good tutorials as well. There is also a  http://www.eukhost.com/web-hosting/ forum where you can find some helpful people.  The people on the invoicing side seem a bit less sympathetic.

You’ll also need to register a domain name but you can probably do that through your hosting company - it shouldn't be expensive.

Setting up Teflpedia

If you want to follow the (sometimes embarrassing) trials and tribulations of setting up Teflpedia read these threads:

http://www.eukhost.com/forums/f28/hello-bob-m-4975/  - Saying "hello" asking newbie questions

http://www.eukhost.com/forums/f11/wiki-hosting-4937/ - More newbie questions

http://www.eukhost.com/forums/f27/created-wiki-but-do-i-need-another-nam... - Questions about naming and URL's
 
http://www.eukhost.com/forums/f16/control-panel-get-into-wiki-4983/ - More technical questions about mediawiki

Anybody glancing at those threads will understand how much you need to work out if you're starting from scratch. 

Technical knowledge

If you want to run mediawiki you need to be prepared to open up the code a bit and do things to it.  This is the only way to install mediawiki extensions and you will want to do that.   It’s not exactly difficult, but editing code is a bit hair-raising if you’re not used to it.

Some wikis are highly customised and have specially written extensions.  Others - like Teflpedia - are more barebones.  If you know next to nothing about wikis then you would be well advised to find some friends with some technical knowledge to help you out.  Teflpedia has been lucky in that regard.