I use a program called
Goldwave for most of my sound-effect needs. It's got a lot of nice effects and all that jazz, but (one of) the really nice thing(s) is that it has an "Expression Evaluator" where you just type in an expression like ((sin(440*t*2*pi)+sin(480*t*2*pi))/8.93) and it will create a wav from that (That last expression was the sound of a telephone ring). You can get some pretty interesting and cool sounds just from that. Then you add some effects to make them sound even better.
Of course, the problem with that is that the sounds don't sound very organic. For that you really do need a microphone out in the real world. You just have to experiment dropping things onto/into other things, smacking things, smacking things against other things, breaking things (use wisely), and generally being mean to household (inanimate) objects.
For example, If hold a CD vertically above your keyboard and drop it just right so it gets stuck in the keys standing up, it makes a great "Thwack" sound. Then take a long stick (any long stick will do), and swing it through the air to get a swish sound. Now combine the two, and you get a convincing Arrow hitting a target (or tree) sound effect.
Or here's another one. Record yourself pretending to eat something like a peanutbutter sandwich. Something that makes your toung make noise as you go through it. Apply a flange to it and lower the pitch, and you've got a Rancor
!
