Having looked at the RPi, it seems like a fairly secure device out the box, as long as you do a couple of things.
The default user/pass needs changed. At the very least, change the password. For better security again, change the username as well. (Add a new user, then disable PI. Check that ROOT is also disabled from SSH login, though I think it is by default anyway.)
Scanning the RPi returns only one open port, 22, which is the SSH connection, and even that has to be turned on before it shows (though most people will be using it instead of a monitor, keyboard and mouse, I expect, especially on a {web}server)
You could change the SSH port number, but that won't do much, since it can be port scanned easily enough. Instead, enable SSH-Key authentication.
You now have no way for anyone to get into your machine without the correct SSH key, username, and password.
Next, set up your webserver. Apache is pretty much where it is at. That will sit and monitor port 80 as default, and automatically response to connections from browsers, serving your webpages.
If you have a firewall or router, you could change the RPi ports, and have the router direct the traffic from one port to the other. For example, port 80 traffic into the router is redirected to port 75 on the RPi, and SSH on 22 is redirected to port 72. This would add another layer of protection, but is a little more complex.
Keep everything updated and patched, obviously.
This won't protect you from attacks that exploit java, flash, SQL servers, etc that you might well add on later, but that's it for the basics, really.
You could also add a firewall, which will slow down anyone who gets into your system from getting out on a different port if they install a new service. Your router should be dealing with that, but if it is directly connected, then set it up, and, for how long it takes, you might as well be running it anyway - it won't add much in the way of system resources.
One other thing you might want to add is fail2ban (http://www.fail2ban.org/wiki/index.php/Main_Page) which adds a firewall rule to block multiple log-in attempts, preventing dictionary attacks. Although these can't work on your system if you've followed the above, if you for some reason need to leave password only SSH auth in place (you remote login from many different machines, for example) then it will prevent a dictionary attack from working. After the number of attempts you specify, it will block for a time any more attempts from that IP address. (Just take care that it doesn't see any router/local IP address and ban that too early or for too long!)
Edited to add: Once you've got everything set up nicely, use a tool like dd or Win32DiskImager to take a complete bit-wise backup of your SD card. That way, if anything goes wrong, you can restore it to the same card or write it to a new card, and carry on regardless. (But if hacked, you'd want to work out what hole was found and close that first, perhaps!)