I've looked into what type of visa I could get. I don't think I have claim to UK/EU citizenship (I think that my ancestors are from England..that counts right? 
) and I'm not married to anyone...I don't even know anybody from the UK.
Unfortunately, your ancestors being from England won't count or help at all. The only thing that would help would be if either of your parents was British by birth and was able to pass their UK citizenship down to you, or if you were a commonwealth citizen (i.e. Canadian) and your grandparent(s) were British.
Alternatively, if either of your parents (or possibly grandparents) were from a different EU country, you might be able to qualify for citizenship of that country.
I could get a company to sponsor me if I start applying for positions in the UK. I've heard it's extremely difficult because the company has to prove there is nobody in the EU that can do the job.
Yes, it can be very difficult - it depends what industry you work in and whether or not that job is considered 'in shortage'.
If the job is on this shortage occupation list, then you can get a visa without them having to prove that no one else in the EU can do the job:
https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/308513/shortageoccupationlistapril14.pdfHowever, if the job is not listed on there, and generally it's mainly high-level jobs that are (one that require many years of experience), then you have to pass the 'Resident Labour Market test' where the company has to advertise the job across the UK and EU for a certain amount of time and can only hire you if they cannot find anyone else who can do the job.
Alternatively, another option might be to get a Tier 2 Intra-Company Transfer visa to the UK branch of your US company, but your job has to be at a certain level and meet certain criteria in order for you to qualify.
I could get a student visa. I have a bachelor's degree from a U.S. University. I'm in a precarious position because I have a bachelor's degree in one industry but have thought about changing to another. I guess I could get a degree in the UK in the industry I want...
You could do, although you would need to decide if it's worth the possible extra cost and whether the courses that are available are suited to you. It will usually only take 3 years for a UK bachelors degree, or 1 year for a masters (if you were able to get a masters in a field close to what you want to do, rather than start a whole new bachelors), so it's a bit shorter than in the US.
Also, if you have a degree from a UK university, and you switch to a Tier 2 work visa while still in the UK on a student visa, you will be exempt from the Resident Labour Market test and so it would be easier to get work sponsorship than without a UK degree.
There's a lot to think about! I wish they allowed job searches when you're visiting. I could come over and search for a job then.
They do allow job searches as a visitor - you can look for jobs and you can interview for jobs as a visitor, but you CANNOT start working for a company until you have returned to the US and been granted a work visa.
But you do have to be careful about this because if you show up at the UK border with your resume/CV and qualifications, UK immigration may suspect you of trying to live and work in the UK illegally.
So, the better thing to do first would be to look for UK jobs online from the US, contact the companies, apply for positions, maybe set up Skype/phone interviews, and then perhaps make a short visit to actually meet/interview in person... so you can show clear plans of just coming to the UK to interview and not that you look like you might try to work illegally.
