Hello :) My DH, DD, and I moved to the UK (northen Scotland) about 6 months ago.
To start most companies do offer some kind of relocation package when moving employees internationally and will help out with moving logistics-like what to do with your US house, cars, etc.
My biggest piece of advice would be to look into joining an expat community here in the UK. They can be so helpful before, during, and after your move here and provide you with expertise for your area. (I belong to one in Aberdeen)
In regards to what to move over, it depends on what your DH's company is willing to ship. Keeping in mind that all your small appliances/electrical things will not work here unless you purchase adaptors. We bought several electric converter adaptors in the States that range in wattage allowance. We have a couple in the kitchen and some in the study/guest bedroom so we can use our stand mixer, crockpot, DH's Xbox, laptop, Ipad, etc.
DH's company allowed a sea shipment (took about 8 weeks) and an air shipment (3 weeks) We ended up bringing everything...which we now regret because things we do not use-lamps, vacuum, coffee pot, curtains/rods....are just taking up storage space in our garage. (alot of the homes here are beautiful, but older and lack closets)
Like PP said, make sure you bring important documents and have your LO's health and vaccination records! It made it so much easier when finding a GP (doctor) here.
Our experience with the NHS has been very positive so far. We registered with a GP (you will need passports, proof of residence like utility bill, and LO's immunization records). Our home health visitor is awesome and very helpful. We even had an emergency while visiting Edinburgh and had to take our DD to the children's hospital after she had an allergic reaction. The care we received was fast and excellent. We already have a follow up appointment with our GP Friday too (this episode happened last week) We plan on trying soon for number two and I have utmost confidence in the healthcare here.
We too brought our dog over here. They recently changed the laws, so as long as your pet is microchiped and has a rabies vaccination there is no longer any quarrantine period.
There is probably a lot I am leaving out. If you have any other more specific questions, let me know!
Good luck! We absolutely love living here in the UK.