You have to make a distinction between the Schengen area on the one hand and the UK, Ireland, Cyprus, Romania, Bulgaria and Croatia on the other hand. All EU countries should grant your wife the right to travel together with you (but not alone) with minimal formalities but those that are not in the Schengen area might still require her to get a separate visa.
For the Schengen area, you only need one visa. So if you want to go to Switzerland (technically Switzerland is not in the EU but for this purpose, it's pretty much the same for the time being), you would apply to the Swiss consulate. With the visa issued by the Swiss consulate you will also be able to travel in most of the EU without any other formalities (but, again, not to the UK, Ireland, etc.)
If you really want to visit all non-Schengen EU countries, then yes, you might need up to 5 or 6 visas (although Romania, Bulgaria, Croatia and Cyprus have some special rules for Schengen visa holders so it might be possible to go there with your wife's Schengen visa as well).
Furthermore, being married should in principle make things easier (at least in the EU proper, not entirely sure about Switzerland, see comments). In your quote, “these” in “check what these are” should be something like a marriage certificate, not the whole documentation package (insurance & financials & itinerary) that other applicants have to provide and the consulates have very little discretion in deciding to issue the visa or not. The visa should also be free of charge (regular Schengen visa fees are currently EUR 60).
Beware, however, all those rules only apply if you are travelling together as they are derived from your right, as EU citizen, to travel freely in the EU. Note that, since you are a UK citizen, these rules also do not generally apply to the UK. Going there might even be more difficult, depending on UK law.