1) Should I then go for applying Marriage Visa or Fiancé Visa?
Neither - a marriage visa is a fiance visa... the official name for a fiance visa is a 'marriage visa' (it costs £826 and is for people who wish to marry in the UK, then apply to stay without going back to the US).
However, if you wish to marry in the UK, then return to the US, you will need a 'marriage visitor visa' (cost £78, allows you to marry in the UK, but you must leave within 6 months and return to the US).
2) If on Fiancé visa and post marriage applied for spouse visa from UK (will use the one day service), can I go back to US and then come back after 2-3 months?
Yes, you can, but you need to be able to prove that you have lived together in the UK for the entire length of your spousal visa, so you would have to have a good reason for being apart for those 2-3 months (and prove it on your next visa application).
You'll need to decide which visa route you would prefer to take:
1) Marriage visitor visa (£78), marry in the UK as a visitor, return to the US for a few months, then apply for a spousal visa (£826) to move to the UK permanently and have permission to work right away.
Total cost: £78 + £826 + 3 flights
2) Fiance visa (£826), move to the UK and marry there, but you cannot work or study until after you have your FLR(M) visa (spouse visa applied for in the UK). The FLR(M) visa costs £867 if you do the in-person one day service.
Total cost: £826 + £867 + 1 flight