Yes, browsers do cache CSS and JavaScript. Toomanyairmiles has already shown you one way to set the cache lifetime of CSS / JS files in an Apache .htaccess file; another way to do it is to use mod_expires:
ExpiresActive On
ExpiresByType text/javascript "access plus 1 week"
ExpiresByType text/css "access plus 1 week"
This tells browsers to cache any CSS / JS files for up to one week — that's long enough to keep requests for such files to a minimum, but still short enough that you won't have to worry about someone's browser retaining an old style sheet from last year in its cache.
(Of course, this only works if your server is using the correct MIME types for these files, but if not, that's a problem which you should fix anyway.)
Now that you've set a long cache lifetime for your static files, the next problem you'll face is getting them refreshed quickly when you do change them (as you almost surely eventually will). A common solution is to append a dummy query string containing a revision number to the URLs, like this:
<link rel="stylesheet" type="text/css" href="/static/style.css?v001" />
The server will ignore the query string (since the URL points to a static file), but the browser will treat it as part of the URL. That way, after you've updated the style sheet, you just need to remember to change the version number to get browsers to load the new style sheet instead of the old. (If you want to get really fancy, you can use a script to generate the version number automatically based on the last modification timestamp of the file.)
In fact, the StackExchange software on this site uses something very much like this trick. If you look at the source of this page, you'll see that the main stylesheet URL looks like this:
http://cdn.sstatic.net/webmasters/all.css?v=e42c8b75839a
Here, the ?v=e42c8b75839a is presumably a version identifier which changes whenever the style sheet is updated. Looking at the HTTP headers e.g. in Firebug, you can also see that the style sheet has a one week cache lifetime (max-age=604800) like I suggested above.