On debian, you just need to apt-get install postfix (I'm partial to postfix). Sounds like a simple install, you won't need any fancy pants mysql backend for that. For just a few users, this is not so bad a way to do things, but for many users, you'll want to start leveraging those complex installs you likely saw that include postfixadmin or some such vdomain manager, that helps with creating accounts. Unfortunately, it's actually easier if you'd accept having unix accounts for each mailbox. Remember, you can set such email accounts to the nologin shell to prevent ssh logins. Below, I have my quick and easy postfix with unix accounts. If you want a basic setup with virtual users too (users with no unix accounts on the box), follow postfix's own howto, it's really quite basic:
http://www.postfix.org/VIRTUAL_README.html#in_virtual_other
So, apt-get install postfix....
During the install, choose internet site as the type of configuration, and answer all questions accordingly, with special attention to "What other destination to accept mail for" I know it sounds counter-intuitive, but put only localhost here for now.
Now you setup postfix for multiple domains:
Create a file you'll use to tell postfix what local domains are. I'm a BSD guy in actual fact, so I like something like in /usr/local/etc/postfix/vdomains, but you can probably go with /etc/postfix/vdomains, or wherever the remainder of your config files will be. Regardless, it's in this directory you'll create a file that lists each domain:
site1.com
site2.org
blah.net
Next, create another file in the same location, say /usr/local/etc/postfix/vdomains/addresses :
site1.com DOMAIN
me@site1.com username
user2@site1.com username2
site2.org DOMAIN
you@site2.org you
etc... (don't forget you can have @site.com to signify a catch-all, and you should have the standard email aliases for postmaster, abuse et al.. see RFC2142
The main points from here relevant to this type of setup is to have the following in main.cf:
mydestination = $myhostname, /path/to/file/with/domains
virtual_maps = hash:/path/to/file/with/addresses
of course there's other standard stuff you should easily be able to find on the intarwebs relative to how to setup postfix, no need to reinvent the wheel here.
Once you're done, run postmap /path/to/file/with/addresses and then postfix reload