Hopefully you know that you need to secure a visa prior to moving to the UK. As someone with a US passport, you can visit the UK whenever, as long as you don’t stay for longer than six months, and provided you aren’t making money (I’m generalizing here; I’m not an attorney and rules change, so always check).
If you’re a student, the information you need is on the US Embassy site. When I had my student visa, I believe it allowed me to work during school breaks, but only limited hours. Additionally, while I was allowed to make use of the NHS (National Health Service), I was not entitled to ‘benefits,’ meaning, for example, I couldn’t collect government income assistance.
Previous versions of the student visa allowed graduates to remain in the UK for up to two years after while working, although those two years didn’t count towards the number needed to apply for permanent residency. That benefit, sadly, no longer remains.
If you are applying for a student visa, it’s easy enough to do on your own. You need certain documentation and must follow all steps exactly, but generally speaking it’s pretty straightforward. If you have an offer at a legitimate institution of higher education, and aren’t otherwise disqualified (e.g you aren’t a convicted felon), it should be relatively simple.
Moving to the UK for work is another thing entirely. Generally speaking, the UK will issue a visa if a) you’re super wealthy and want to invest money in starting a company in the UK or b) you are a skilled or highly skilled worker in a field that has a dearth of potential employees.
The details of the steps involved in securing a work visa can be found on the UK government site.
In our case, my partner Austin applied for and was offered a position at a small tech company. His particular tech field is not as large an industry in the UK as in, say, Seattle, so they don’t have as large of a pool of potential employees to choose from.
I strongly recommend that, as part of your negotiations with your new company, you insist that they pay for both the visas and the visa process, as well as contact with an immigration attorney with experience in this field. There are a lot of ways to screw it up, and immigration attorneys do this for a living. It’ll remove a little bit of the stress from the process.