You can find the exact cost on the British consulate's website, but I believe the cost is hovering around $500 right now. You will need to apply from the US and you can get the visa in a matter of hours if you apply in person. If you apply by post, the waiting time seems to be anywhere from 2 to 4 weeks.
Alternatively, and I'm not suggesting this--though there are a couple of people on the site who've done it, you can get married in the UK on a visitor's visa. You'll just need to go back to the US to apply for your spousal visa, which, again, you can get in a couple of hours if you apply in person.
UNLESS you're currently here on a student visa worth 6 months or more. In that case, you can marry here and adjust your status to spouse without having to leave the UK.