The UK Government website has the form and the guidance books
here. I recommend that you read the form and both booklets thoroughly before filling in the form, as it is likely that any questions you have will be answered by the information they contain. As an EEA citizen, you will probably be expected to show proof of living in the UK for the last six years. The guides will tell you the documents that are admissible.
The form to pay the fee (along with a list of fees) is the last page of
this document.
Finally, it is the general view of people on this forum that you are best off applying through a
Nationality Checking Service (NCS) that has a Joint Citizenship and Passport (JCAP) service. For a fee (usually £80, I believe), they will check your application and supporting documents. They will send certified copies of all your supporting documents (except for the Life in the UK test letter) to the Home Office, which allows you to keep the originals. This means you can travel while your application is under consideration. This is in contrast to postal applications, where you must send originals of all your documents.
However, using an NCS does not guarantee that your application will be approved, it does not expedite the amount of time to a decision, and it does not guarantee that the Home Office won't get in touch requesting that you send original documents or more information (though the last one is rare and generally doesn't happen to people who follow the advice of their NCS caseworker).
You do not have to use your own council's NCS service (sometimes your local council doesn't have many available appointments or doesn't offer a JCAP, but a nearby council does), though if your application is approved, you will be expected to attend a citizenship ceremony in the local authority that you live in (if you want your ceremony to take place elsewhere, you can specify that in section 1.21 of Form AN).
