National Insurance Number -
http://talk.uk-yankee.com/index.php?topic=3119.0NHS - Look up a surgery in your area, ask if they are taking new patients and go in to register. Sometimes you have to bring your passport and proof of address, but not always.
http://talk.uk-yankee.com/index.php?topic=47774.0Insurance - You just need to start asking around. Some do, some don't.
Your work will tell you what you need to bring on your first day. Typically, it is just your passport.
You don't register for perm residency, citizenship. YOu have to complete applications and meet the requirements just like you did for your spousal visa. Have a look on the visa pages as there is too much for me to re-write here. It is also going through a transition in the rules and no one quite knows what is going to happen. In the meantime, you can at least get familiar with the process as a whole.
For taxes, this is all operated through your employer and most people don't have to file a return. You need to complete Form P86 and you will get a tax code that indicates your taxability. Review that when you get it.
YOu can drive for 1 year on your US license. You may have difficulty getting insurance cover with a US license. YOu will just need to call and ask around.