Is extremely slow...

Firefox and IE. Anything involving the is slow. Im
using an auto_complete field. When you begin to type after 3 characters
it should generate a list of options. In order for it to work you have
to stop typing because it will never keep up with you. I have to wait
about 4 seconds in order to see a populated list.

This almost certainly has nothing to do with the server since it's all
javascript code running in the browser at this point. How much RAM
does the client machine have?

I don't know about IE, but every version of Firefox I've used so far
seems to have one or more memory leaks which means that Firefox slows
down over time, killing Firefox and restarting it often improves the
performance dramatically, at least for a while.

It seems to be getting better with each release of Firefox but...

I am also using some of the AJAX stuff like remote_function so I do not
have to reload the page every time. This is extremely slow too. I have
made some clickable areas on the screen and when a user clicks on one I
use replace_html inside the div. This again takes about 5 seconds to
see a response.

Some of this might be a server problem, but I'm guessing that if
you're seeing the kind of performance you are for local javascript
execution, that it's the same story.