Estimated Costs of Moving to England, United Kingdom

Yes getting your visa is the main thing. Once you are living here, working, and paying tax, medical is basically free (hope it's still like that).

Apartment cost varies enormously depending on location and region. What you could do is google to some property websites for example Zoopla which does rentals as well as property sales. I use them just to keep tabs on property prices.

Transportation costs, it's better to bring as little as possible with you. Try to bring everything in your luggage. Anything else can easily be found. The electrical supply in the UK is not 120V like in the USA but 230V so most electrical items are just not worth bringing here as you'd need to buy a step down transformer. However, items like cell phones and laptops usually work on any electrical supply.

If you still want to bring lots of stuff over, then you need to contact a cargo company or freight forwarder in the States and get them to quote you per cubic meter and a total cost of everything for door to door delivery.