Most used capacitor values
The values/types will somewhat depend on what kinds of stuff you are designing, but here's an attempt at an answer:

Decoupling:
100nF ceramic, lots and lots of. 
Also now ceramic caps have improved and become cheaper, you can use them to replace electrolytics up to ~10uF.
Mouser do some real cheap large ceramic values.

For lowish speed (say up to 40MHz) the above should do fine. For higher (e.g. fast micros, FPGAs, etc) you would need:
1uF ceramic - lots of.
(100nF ceramic)
10nF ceramic - lots of.
Possibly 1nF ceramic lots of too, depending on the manufacturers advice.

Crystal bypass caps:
15pF, 20pF, 30pF ceramic. For typical microcontroller crystal oscillator (or a discrete inverting gate version) 

Electrolytics:
Grab a few in the range from 100uF to 1000uF, make sure the caps voltage rating is suitable for your circuit. Double the rating is good (e.g. if circuit runs at 5V then use >10V rated)
For audio you may want a quite a few 10uF, 22uF, etc tantalum/Al and also some non-polarised electrolytics. If working with power amp output stages then high quality e.g. 1000uF electros may well be needed, depending on power supply (single/dual rail)

Plastic film, Mica, etc:
If you are working with sensitive audio/RF analog circuits then you will likely need a decent range of stable/good quality capacitors. Probably best to order these as required.

If I think of any more I will add them..

Other caps I would play by ear mostly. You will almost certainly end up using a few of each value ceramic from 10pF up to 1uF if you are busy, but since you can very quick delivery nowadays you could simply see what you use most (e.g I get next day delivery with an order up to 8pm) You will quickly build up a good range either way.