The ARRL will also do the renewal for you if you are an ARRL member. They will send you a notice in the mail with simple form that you fill out and return. No cost to members.
Note that you cannot renew your license until 90 days before it will expire. There is no cost for renewal.
First, you need to get an FRN Federal Registration Number or something like that. The easiest way to do that is online. Go to wireless.fcc.gov/uls and select the CORES registration link. Follow the process and you will have that part of it over with.
If you would rather get an FRN by mail, download the Form 160.
Second step is the renewal itself. You can do this online too (back to the link above). But you have to download several Mb of java stuff to do it. My preference is to download the Form 605 which includes the instructions (at least 10 pages!).
Fill in the form and mail it in. You will have to "certify" that your station is not "likely to have a significant environmental effect" according to Section 97.13(a) and that you have "READ and WILL COMPLY WITH Section 97.13(c)", which is basically that you aren't going to fry anyone with rf. The ARRL has all of Section 97 in three formats: HTML, PDF, or plain text here.
If you need to change your address, print two Form 605's. Use one to change your address, mail it in, wait a week, and use the second to renew.
Questions? Email me if you would like, but this is about all I know about this.
73, de wb0taq