Copenhagen is only a short hour and half [ish] flight from London and is served by the usual range of budget airlines, so getting there is easy peasy and inexpensive. It is an expensive city though and I had to do a fair amount of research to keep costs reasonable.
The S- train departs every 12 minutes from the airport and is modern and reliable which is not something we are used to in the UK! It takes just under 15 minutes to get to Central Station in Copenhagen from the airport. This was ideal for us as our hotel was only a few minutes walk away. Other options are the Metro but this does not serve the Central Station.
I have split Copenhagen into two convenient sections for this website [or is it a blog?] – Central Copenhagen which covers the districts of Tivoli, Strøget and Slotsholmen and Outskirts which covers Nørreport, Nyhavn, Christianshavn and Vesterbro.
Most of the main places that I recommend you visit are in or just outside of the city centre so you can either walk there or use public transport. The Metro, S- Train and busses are plentiful and efficient. There are bicycles everywhere!
The yellow Movia Harbour Busses were brilliant and a wonderful way to see the city from the river.
Hotels in Copenhagen were pretty pricey but I managed to find one which seemed reasonable and had favourable reviews. It was located in the red light district behind Central Station, so it was very convenient, but I was approached by very friendly young ladies a fair amount despite being with my wife! The Saga Hotel was comfortable and breakfast was fine. Airbnb would also be an option worth considering if you are on a budget.
The city is a mix of classic, historic buildings and contempory modern architecture which makes wandering around the streets a delight, you never know what you may discover around the corner.