No no I won't go!
A lot of people we have met during our stay here came "like us" for a couple of years. Twenty years or so later, they're still here. As a British citizen you don't need to do much in order to stay here long term. As long as you've done all the red tape, got your insurance sorted and you remember to change your address at the Einwohnermeldeamt if you move, that's it. You can stay for as long as you want, provided that is you can support yourself.
Although as a member of an EU country you have unrestricted freedom of movement in the Federal Republic of Germany and do not require a work permit. You do need to carry out the following legal obligations:-
Everyone who lives in Germany, regardless of nationality, must be registered with the local authorities, in Germany, the address of every resident is recorded. When you arrive in Germany and have found accommodation you need to register with your local registry office (Einwohnermeldeamt / Meldestelle), you must do this within one week of arrival. All dependants must also register in person.
If you stay in lodgings or a hotel you still need to register if you are there for over two months.
The registration form (Anmeldeformular), which can be found in any stationery store, must be signed by you and the owner of the property you are renting. If you own the property take proof of ownership.
When you arrive at the registration office, pull a number and wait. Take "War and Peace" with you if you live in a highly-populated district and do not like getting up at the crack of dawn! On completion of the registration process you will receive a registration certificate (Meldebescheinigung / Anmeldebestätigung ) a document which proves that you are registered at your address - keep it safe! You will need this piece of paper again and again as you start to open bank accounts, apply for residence permits, join libraries etc.
You need to repeat this process every time you move, even if you only move within the same city, and if you move to another city for goodness sake don't forget to check if you need to deregister before you move Munich-Frankfurt-Munich-Frankfurt is a lot of kilometers!
You need to take your passport (Reisepaß) and the name and address of your landlord. In order to expedite registration take your rental contract (Mietvertrag) and make sure your employer gives you a contract or letter of confirmation (Bestätigungsbrief) of your employment. Making photocopies first and offering them to the clerk always goes down well, but make sure you have the originals too.
As an EU citizen you may enter Germany without a visa or work permit, you can also take your spouse and children (up to the age of 21) with you, however you do need a current 10 year passport.
As a happy little citizen of the EU you have an almost absolute right to be in Germany, you do not need a resident's permit (Aufenhaltserlaubnis), but the German authorities like pieces of paper, it's nothing personal, even Germans have to carry an id card (once you've made some friends take a look at them - they are really funny). So, you will need to get a "Bescheinigung über das Aufenthaltsrecht". You can get this piece of paper from the "Einwohnemeldeamt" or "Auslanderamt" in your area.
If you stay here long enough, you'll need to renew your British passport. Passports are biometric now and so it all seems to be a bit more stressful, especially getting the photo just right, somehow the rules for German passports are slightly different, I recommend using a professional photographer, it costs a bit more than the photo booths, but it's also quite fun (I still ended up looking like a criminal though).
Make sure you have all the documents you need to prove you are you and any documents you need for any children you have.
You can print out and fill in a passport application form here. The British Consulate in Dusseldorf handles all the paperwork and they can also provide emergency travel documents etc. Of course it will all cost you; being British is expensive.
Hopefully you'll have learnt a bit of German before coming here, but if you're going to stay you'll need to take it a bit more seriously. Doing it for fun is one thing, doing it to get a better job... Well it can be a bit too much like hard work. There's some advice on how to learn German here.
One depressing thing to think about is that if you are staying here, you may end up dying here, not so much living in Germany as dying in Germany. This raises new questions. Will you want to be buried here? Does your will comply with German law? I know you won't get the headache if things aren't in order, but your executor and heirs will thank you if you simplify things. In some cases (for example if you have property here and in the UK) it might make sense to have two wills.