Personally, I'd recommend marrying in the US and applying for the spousal visa, especially if you are only going to have a civil ceremony. It's easier and cheaper since you only have to apply for one visa. The fiance visa is good for people who want to have a big wedding in the UK, or whose British fiances can't travel to the US for some reason. Also, with the spousal visa you are eligible to work as soon as you arrive in the UK, whereas you cannot work as a fiance.
It's very unlikely that you will mess up the process, whatever you choose. As long as you read the requirements carefully and make sure that you meet them, you'll be fine. This site is a great resource. Good luck!