Budget US$75 per day for normal meals and minor incidental expenses. Does not include alcohol or better restaurants. You can decrease it to $60 per day if breakfast is included in your hotel bill. You may be able to get by on less, but don't budget it that way. You may also have to pay for on-board meals when flying between cities.
The answer from Willeke is correct and you can mainly use public transit. I have not been to Copenhagen, but taxi rates in Amsterdam, Belfast, and London are more expensive than in most US cities.
Remember to budget for transportation between airport and hotel.
Use travel guides to estimate costs for activities.
I normally do not book excursions (ex. to Stonehenge) very far in advance. It's not necessary, and waiting until a few days before may let you choose a better-weather day.
Check with your health insurance to find out if it covers you outside the USA and pays for medical evacuation to home (should that be necessary). If yes then take proof of coverage with you. If no or not sure then get trip medical insurance, which is cheap and sold by airlines & travel agents. PLEASE do not skimp on this as an otherwise-silly accident could turn into a crisis if you don't have medical insurance.
http://travel.state.gov/travel/cis_pa_tw...
In non of those cities you need to use taxis. If you do, your budget for just taxis can be more than your budget for all the rest of your expenses. There is good public transport, with trains, subways and buses and trams.
Most hotels have public transport information on their sites, with the nearest underground, bus or tram stop, (which one depends on the city) and often a simple 'how to get there from the main airport.
Just sight seeing, modest eating and a few main sights or museums per day should set you back
