You can ride for 12mths on the visitor scheme on your US licence as your US licence holds the equivalent of a motorcycles category (as long as your US licence is in date and valid).
I'd recommend applying for your UK provisional licence ASAP, this will give you a provisional category for motorcycles, cars and other vehicles.
Then I would say you need to make a decision what to go for first. Roughly speaking adding a category (be it motorcycle, minibus or lorry) requires a theory test and a practical test...and lessons somewhere in the middle. So to add cars & bikes you would need the car theory, practical and then the motorcycle theory and practical - which is not cheap all at once.
If you want to go for bikes first and you are over 21 then you can take the DSA direct access scheme if you take your bike test on a machine over 50hp. This then allows you to ride any size bike if you pass your test.
Be warned when you pass your bike test, if you wanted to drive a car you would need a supervisor and 'L' plates as you would only hold a provisional entitlement for a car until you passed a test in a car.
As for cost, I would say your not far off £1K...depends how many lessons you need and whether you want to drive a car as well.
