Download the precompiled installer (or the source and build the installer with ant build)
Run the installer via java -jar install.jar
Answer the prompts:
1. Specify an I2P installation directory (e.g. C:\i2p\ or ~/i2p)
2. Provide your publicly reachable hostname or IP address - if you have a NAT or firewall, be sure to use the external address forward the I2P network port (default 8887) to your machine so that you can be contacted
3. When asked whether you want to reseed, if you say yes it will download a set of references to existing routers from http://dev.i2p.net/i2pdb/. If you say no, you must get references from someone else running I2P.
Download the precompiled jbigi library for your OS and CPU to significantly improve the speed and overhead of running I2P. Save this jbigi.dll or libjbigi.so into your I2P installation directory.
Start up the router with the startRouter script located in your I2P installation directory
After two minutes or so, there should be an HTTP proxy listening on port 4444 and an IRC proxy listening on port 6668. You can set your web browser's proxy to localhost:4444 and browse to any eepsite or normal website through it (if you go to http://whatismyip.com/ it will come back with the IP address of the outproxy - 69.55.226.145). You can also fire up your IRC client (e.g. irssi or mIRC) and tell it to use the IRC server at localhost:6668 (which will connect you to irc.duck.i2p).

If you are unable to reach those ports on your machine or cannot reach any sites or get on IRC, please check your router console and make sure that there are TCP connections listed (instead of a big red warning). Typical problems include having the wrong port forwarded, entering the wrong hostname/IP address during install, or not being able to use NTP to get the right time (which requires UDP port 123).

If those are not the problem and you are still unable to get any TCP connections, or if there are problems using the network, please jump on IRC at irc.freenode.net/#i2p and we'll see what we can do.