A fiance visa must be applied for from outside the UK and the fee as of April 2015 is £956. The earliest you can apply is 3 months before you wish to travel to the UK. You can ask for the visa to be post-dated by up to 3 months (the application will ask for your travel date and the date you put will be when the visa is made valid from - this date cannot be more than 3 months ahead of the date you apply online).

1. Gather up all your documents - financial requirement documents, accommodation requirement documents, relationship evidence, passports, letter from sponsor (your fiance), evidence of plans to marry etc.
2. Complete the online application form, which can be found here:
   https://www.visa4uk.fco.gov.uk/ Submit the application, pay for it, book biometrics appointment and print the application and biometrics confirmation sheet.
3. Print and complete Appendix 2 of https://www.gov.uk/government/publications/application-for-uk-visa-for-f...
4. Attend the biometrics appointment, taking your passport and biometrics confirmation sheet. They will stamp the sheet at the appointment.
5. Place your passport, the stamped confirmation sheet, your application, Appendix 2 and all your documents (and copies of everything) in an envelope/package. Create an online account with FedEx/UPS and set up an international return waybill for return of your documents. Also include suitable return packaging for your visa and documents. 
6. If you want, you can purchase Settlement Priority Processing from the Worldbridge website, printing the receipt, putting it in the package and writing 'Settlement Priority Service' on the outside.
7. Send your documents to Sheffield, UK for processing within 14 calendar days of the biometrics. (and within 30 days of applying online - whichever is sooner)
8. Wait for the visa to be processed and mailed back to you. (1-4 weeks, processing times vary)

95-98% of fiancé visas are approved for applications made in the USA. If you meet the requirements and included all required documents, there is no reason it would be refused.

After you arrive, in England, you must wait until you have been present for 7 days and then you can go to a designated register office to give notice to marry. The notice must be displayed for 15 days before you can marry. If marrying in Scotland, there is no residency period.

After the wedding, you can apply for Further Leave to Remain, FLR(M) extension as the spouse of a UK citizen (£649). 

Alternatively, your fiance can fly to the US as a visitor, you can marry in the US and then just apply directly for the spousal visa. That would mean only applying for 1 visa instead of two and you could then work immediately in the UK (you cannot work at all, even unpaid, on a fiance visa or a visit for marriage visa).