There's no way to enter the UK without a visa. The General Visitor (tourist) visa stamp you get when you come to visit is still a visa. On the General Visitor visa you can't get married or register to marry:
from
http://www.ukba.homeoffice.gov.uk/visitingtheuk/visitors/eligibility/ou must also show that, during your visit, you do not intend to:
take employment, produce goods or provide services, including the selling of goods or services directo to members of the public;
undertake a course of study;
marry or form a civil partnership, or give notice of marriage or civil partnership;
carry out the activities of a business visitor, a sports visitor or an entertainer visitor; or
receive private medical treatment.
The Church of England thing had to do with the Certificate of Approval (CoA) which is now defunct. People who were here on a visa other than General Visitor or one that granted specific permission to marry (Marriage Visitor Visa or Fiancé(e) Visa) had to fill out a CoA, requesting permission to get married. The exception to this was if you were having a CoE wedding.
Your options are to:
A) get married in the US, apply for and get a Spousal Visa, move to the UK
B) apply for and get a Marriage Visitor Visa, get married in the UK, return to the US, apply for and get a Spousal Visa, move to the UK
C) apply for and get a Fiancée Visa, move to the UK, get married, apply for and get Further Leave to Remain (Married) Visa.
D) apply for and get a work or school related visa, move to the UK, get married, apply for and get FLR(M).
"A" is usually the cheapest/easiest. "C" is very expensive as the Fiancé(e) Visa costs about as much as a Spousal or FLR(M) Visas. As for "B", the MVV isn't terribly expensive but you have to factor in airfare, though I suppose option "A" has the same airfare costs, only it'll be your fiancé for two of the three flights. "D" really has nothing to do with getting married and is becoming increasingly difficult, but if you're not ready to get married right away and can afford tuition, a Tier 4 (student) visa is probably your best bet.
As for the ceremony itself, a registrar (UK) or courthouse (US) wedding is probably the least expensive option.