What kind of drinks I will have to know (that are quite common) for bartending?!
You've recieved some bum advice so far.
Bartending "schools" are a sham, and myself and every colleague I know, intentionally refuses to hire bar school grads. They learn a few drinks and think they're the shi&, but can't even change a keg, know nothing about wine/pairings, love sour mix for some unknown reason and stand behind a real bar with the deer in the headlights face, praying to God the customer orders a bottled beer.
Also, martinis DON'T come in a thousand different varieties. A martini is made with gin, a good amount of dry vermouth (none of this bone-dry crap: that's just a glass of cold gin), traditionally a splash of orange bitters and a twist, although olives are now acceptable. Anything else is a cocktail, NOT a martini. Recently, vodka martinis have become popular, but again, they're not martinis, they're vodka martinis.
This link has good suggestions: http://www.mixographer.com/2006/05/top-t…
I'd add a Manhattan, a mojito and an Old Fashioned to that list, as well.
Hiring barstaff for 15 years.
You should really take a bartending class, you're going to have to know a ton of drinks. Either that or buy a cocktail book and memorize it.
Martinis come in many, many different mixtures. Good luck with trying to learn them all. Every time I go out, I see a new flavor of one made with interesting ingredients.
Too many to list. Get to the bookstore and get yourself a bartenders guide that tell you how to make everything.
There are way to many to name but I agree with the others a bartending class would be great use to you.