I believe that to have a more or less productive path you should start by adding a separated spi to ethernet IC, such as the ENC28J60 and improve from there. Such chip will abstract OSI layers 1 and 2.
Once you have the hardware I believe that certainly programming will be the most challenging part, as you'll have to understand the details of the IP, TCP and whatever protocol you'll use.
Probably you'd prefer to learn by example before digging into all the theoretical details, so I believe that taking a look at what is made on sites such as tuxgraphics (article on avr web-server, with code - open-source), embeddedmarket (check their user-manual for the schematics). Also check this blog post.
Surely you will need as much RAM and EEPROM as you can, you can do it with a device of the 8bit family.
Also don't forget that you can start with something quite simple (no web-server, just an telnet-like connection, or raw (simply transferring your current serial interface into one based on sockets), and use software such as putty to connect into it).
On a final note, in case you need a quick solution, you can also use a drop-in converter for your serial/ttl communication port. There are versions for consumers (digi one SP, moxa Nport5150, etc) and others to integrate on your project (from moxa, hw-group, and others). These completely abstract the osi layers 1, 2, 3, 4. Well, they abstract everything.
Hope this helps getting you on track.