hi ash... You only need one MTA on your system, I suggest you choose one of postfix or qmail (or exim). actually you CAN only run one, only one at a time can listen on port 25, which it does to be an SMTP server. People have "religious wars" about which MTA is best, but really, unless you're running a huge mailserver or have some unusual requirements, any one will do the job. If you don't have any better reason to choose, pick the one with the best documentation and support.
qmail (and any other MTA) can be a "mail relay" - it means they will accept email messages with SMTP, then pass them on to the right server. This feature doesn't relay POP though.
What you can do though, in the situation you've described, is use Fetchmail to get mail from the server it's on now (like an ISP's POP server?) and deliver it to a user's mailbox on your server. Then the clients on your network can collect it whenever they're ready... you run a POP server (or IMAP or webmail) on your server to allow them to do that. When they send mail, they send it to your server with SMTP... which relays it to the rest of the internet. Is that going the way you want?