The internet is usually the easiest and simplest place to find Sound Effects. This is a selection of the most useful websites that have so far been found. In order to stay in your sound budget (if you have one) it is usually best to first go through all the free Sound FX libraries to fill most of your requirements and then use something like Sounddogs for particularly specific or unusal requirements.
Partners in Rhyme - Probably the best free library, particuarly for Weather and Animal sounds. An excellent first port of call for most shows.
Tintagel - A large library with many effects of varying usefulness and sound quality.
A1 Free Sound effects - Another large library with an esoteric selection of effects. A trawl is usually required but can be rewarding, particularly for cartoon type effects.
Fx Links - A set of links to other free libraries.
Freesound - A website with a growing set of of Creative Commons licensed sounds, some of which are good quality. Many of these are sounds people have grabbed of "Real Life" events e.g. there's a good set from the London Underground. If you've created your own sounds that you're happy for others to use, you can upload them here.
Sounddogs - Sounddogs is an extremely extensive library that can be relied on to provide any sound effect you might need, usually in several versions. Prices are a little on the expensive side (£1-£4 mostly depending on length of effect) so it is easy for the costs to rack up. A reasonable shopping list comes to about £30 usually. Previews are provided so you can audition each sound, an it is worth pointing out the non-obvious fact that you are only given 30s of what may be a much longer track. If only a small section of a large track is needed (e.g. only 1 of the 3 cars passing) then you can specify at checkout which section you want, and are only charged accordingly. Prices are all in dollars as it is an American webiste, and you will need a visa or other credit card to pay (i.e. not Switch/Maestro).
The Sound Effects Library - This is a British alternative to Sounddogs with very similar functionality and pricing.
SoundScalpel - Cheaper effects, in general, than SoundDogs or the Sound Effects Library, although their collection is currently not as large. It seems to be expanding week-on-week, however, and you can preview effects before buying. Once purchased, it is very easy to re-download your effects from a handy "your account" menu, and look out for their "free sounds of the week"!