I don't have a favorite general purpose lens. I have favorite special-purpose lenses.
The trick with any lens is to know its strengths and weaknesses and work within them.
With the kit lens--its main weakness is that it's soft wide open, and it's slow. So, if you use it outdoors, stopped down to f/8 - f/16, that can make up for a great deal of the kit lenses weaknesses.
Its strengths are that it's a) cheap enough to be trashable, so you can be fearless taking it along and sticking it into anything; b) it's small and light so nobody notices you with it and it can go everywhere; and c) it goes 18mm wiiiiide--it's a superb option for landscapes and used close-up for portraits for that reason.
If you're on a really cash-strapped budget, my recommendation for another lens would be the EF 50mm f/1.8 II. ($80). It is, however,
not a "general purpose" lens. It is a portrait lens, designed more for available-light photography (i.e., it can be used indoors without a flash). It's optically better than the EF-S 18-55, but that's easy for it to do because it doesn't zoom. It will teach you how to zoom with your feet, and possibly how to search out other angles of attack. Also, it will teach you how to use your autofocus system.