last time I did this (about 13 years ago) it went like this:

you have to get a fiancee visa before travelling to the UK - it's a quick process
then you go, with your visa
you have 6 months to use it
You can go to the registry office and do a quickie wedding -- that's what we did, it was lovely.
once it's official, you can apply for a leave to remain indefinitely on your passport -- it takes awhile. in the meantime, you can't work in the UK.
if your new spouse chooses to move to the US, they must apply for a greencard BEFORE they can move over. it takes awhile.
I think the reverse is true if you choose to marry in the US and stay here; your fiance must apply for a visa prior to coming over, then you make it official, then they can apply for a greencard.

It's worth figuring out where you want to live before you get married, so that you can streamline the process. It's easier to get a residence visa in the country that you got married in. Greencard apps through the office of homeland insecurity can take for ever; the UK is a bit faster.

Both cost money - so plan on an outlay of cash.

mazel tov!