Perform a search on this site for 'MaxClients' which should turn up quite a bit of information to get you started.
There may be a few directives you want to look at depending on your server resources that could affect the number of simultaneous connections and thus performance of your web server. Here are a few examples (some may or may not need to be configured depending on your environment):
On the Apache level you'll want to maximize the
As a rough guide you can run
If you want to increase MaxClients further you can try to reduce the size of the Apache processes by not loading any module you don't need.
As you mentioned, you may wish to consider an alternative web server, or possibly dual Apache web servers, to increase the number of concurrent connections possible. When Apache is configured to server dynamic pages (PHP, etc...) it is relatively memory hungry which puts a relatively low limit on how many connections you can have before running out of RAM. The usual choice in this case is to use a lighter web server (Lighttpd, Nginx, etc...) to serve all the static content and leave Apache to just serve the dynamic pages.