Questions regarding passports, working/living in UK, etc

First, a little background.  I am a US citizen born and raised in the United States.  Both my parents were born in the UK but are also US citizens and have lived in the US for 40 years+.  I still have a number of extended family members who have lived in the UK all their lives.  I also have a british birth certificate in addition to my US one (thanks to forethought from my mom when I was born).  After reading various pieces of information on websites and in print, it seems I would have no problem obtaining a British passport.  I am planning on moving to the UK sometime in the coming year but am unsure what steps to take to get started. 

Is it advisable for me to get a british passport or a visa? 
What about working... would I need a work visa if I get a british passport? 
Would this affect my US citizenship? 
Would I need to be sponsored by a potential employer?
Would I be able to keep my US passport and are there restrictions/limitations to when I need to use it in order not to jeopardize my US citizenship?  How do I establish residency in the UK... is working/living in the UK with a british passport enough? 
Does having a british passport mean that I inherently have "right of abode"?

I have read various articles on all sorts of websites but it is all a little confusing and a bit of a jumble.  I found this site hoping I could learn from and get advice from other US citizens who have moved to the UK.  Any help or advice would be greatly appreciated.  Thank you for your time 

1. British passport, without a doubt.
2. No visa of any sort needed- your passport gives you the right to live and work in the UK without any restrictions.
3. No.
4. No.
5. Yes- use your US passport when entering and leaving the US, and you'll be fine.
6. Yes.

Rose is spot on. Get the British passport and you can live and work anywhere in Europe.

Fabulous!  Thank you so much... so nice to have an answer so fast.  There is so much information out there it's mind boggling 

Cheers

Does having a british passport mean that I inherently have "right of abode"?

Small technicality:  By what you've written you have ROA now.  A passport allows you to exercise it.

Thank you for the clarification... that makes sense