Don't worry, I will not give the 9876th introduction about TCP/IP here! There are so many good books and there's so much good stuff around in the Web. I will just try to give some advice on how to get familiar with IP.
It would be helpful to have some basic understanding of data communication networks and to have some experience with Unix systems. Otherwise, it will get a little harder to learn TCP/IP, but it's still manageable.
As a starter, you can read RFC-1180, TCP/IP Tutorial, which you can download from the Internet Engineering Task Force (IETF).
Spending some more time, you should also read Comer's book:
This book's the TCP/IP 'bible' and thus a good investment, you should consider to buy it - you'll read it probably more than once.
Also good, but a little less readable than Comer's book, is IBM's TCP/IP Tutorial and Technical Overview. There's also a free downloadable PDF version! Nearly 1000 pages of condensed IP know-how, so go and get it!
After this, it's a good idea to extend your IP knowledge by studying the actual RFCs. Today, IP is such a vast field that you will have to concentrate on the areas that are the most interesting for you. Some practical experience with IP network configuration would also be helpful, unfortunately not everybody has access to routers - perhaps to can play a bit with Linux.
Also, surf the Cisco pages. Cisco's the absolute market leader and innovator in Internet technology, you will find lots of valuable information on their pages. For example, Cisco's Internetworking Technology Overview is a good starting point for the 'absolute beginners' in data communication. Also, don't forget Juniper , currently the runner-up in market share. There's a lot of good stuff (e.g. white papers) on their pages, too.
It will take more than six month of hard work to become a true IP expert, but you can come quite far in that time. It's not too complicated (no math needed ...), it's simply so much and everything is somehow related. Don't give up if first there seems to be no light at the end of the tunnel.
Disclaimer: IP knowledge is not very communicative (sole exception: datacom people). So most probably, you will not be able to impress your girl friend with all that router stuff (although she'll love the money that you can make with it), and advocates and (medical) doctors will surely be the center of attention at a party and NOT you poor IP expert! So don't waste all your precious time on IP (you could), instead also learn something really useful as e.g. playing the guitar, singing, windsurfing, rollerblading and so on.
Some TCP/IP links: