Generally, the NHS works on a 'free at the point of access' basis, i.e. you walk into a NHS hospital or medical centre, and consultation/treatement is free, regardless of whether you work or pay taxes. There is, however, a fixed cost for prescriptions (currently £6 or so) and you have to pay for dental care (either a fixed NHS cost or a cost levied by the dentist). My American wife has NEVER been asked for ID or proof of residency to qualify for treatment and we've even been able to take visiting relatives to see our doctor at no cost. I think we have been lucky and it has certainly helped that I am a UK citizen and we have registered with a GP together.
I think there are rules on eligibility. My simplistic understanding is that if you are here as a temporary resident or from the EU, you are entitled to the same care as UK citizens. You may want to check on the NHS website or with the UK Embassy - there's stuff on this web site too. Like I say, we have never been asked to prove eligibility, but I think the NHS may be 'tightening up' and may check, but I can't see staff asking to check documentation before treatment - the system just isn't like that here.
The key to healthcare here is registering with a GP, who then refers you for treatment. Unlike the US, patients very rarely self refer and if they do, it would have to be on a private basis. So, the most important thing is to get a good GP! When you move, don't rush to register with the nearest GP, but ask your neighbours/colleagues, who they would recommend. If you're not happy with your GP, you can change. This is the theory, but you may find that in practice, it can be tricky as there may be no other choice! We have had some mediocre GPs, now have an excellent one and are generally very happy with our family's medical care.
I also have private healthcare through from my employer, which just means that a referral can be made to a private hospital and you may get seen more quickly. However, my experience of private healthcare so far has been bad - too many forms to fill out to be approved for consultation and then referred to some hospital miles away, when I could have gone to the NHS hospital in my town and probably be seen by the very same doctor!
Anyway, everyone has experiences to share of the NHS, so scan some of the other threads to get a wider picture and good luck with the move!