Common types of diodes to keep around
1N4148 and 1N400x(*). Definitely.  

The 1N4148 is the standard signal diode, the 1N4001 a rectifier capable of 1 A and 50 V. If you need higher voltages you can go for the 1N4002 through 1N4007, for 100 V and 1000 V respectively. See also this answer. 

Zeners. You don't want zeners. :-) Well, you could keep a few, but what voltage(s)? Most often you'll use a three-legged regulator, like an LE33, for instance. They regulate much better than zeners. Unless! (and this should please Russell) There's always the TL431, which is an adjustable zener, and because it has an adjustable voltage you only need one type. Costs hardly more than a zener, but has much better specs.



(*) I first mentioned the 1N4001 here, but on second thought the 1N4007 may be a better choice: you can use that for almost any application, including rectifying 230 V AC, like Olin says. Not what I would need everyday, but the 1N4007 is exactly the same price as the 1N4001 at Digikey anyway (6.49 ¢ a piece @ 100s), and the 1N4007 has a lower junction capacitance as well.