I'm only intimately familiar with the US policies, but most airlines will not permit 3 carry-ons. Your best bet will be to very carefully pack and pad the instrument to avoid damage. Do this a day ahead of time so you have as much time to make it perfect as you need.
A backpack and laptop, as long as they aren't oversize, should not be a problem, but the guitar-sized-instrument will probably not be permitted as carry-on.
Additional luggage is usually only permitted as checked luggage - they probably will not permit additional carry-on, even if you are willing to pay for it. The US has been particularly stringent on this issue lately, with some airlines charging even for the first and second checked bag these days, but generally your fees for additional checked luggage will be high - here in the US it is usually around $50 per bag, with the cost rising steeply with extra bags, and if the bags are overweight. (50 lbs in the US, usually, 25 KG in the UK -- varies by airline)
As for transporting it, that depends on you and your resources. Got anybody who can help you move it? The skycaps here have huge carts for moving luggage, and with a good tip they will practically walk it to the plane for you. If there are airport personnel to help people with their luggage, seek them out when you arrive. If there aren't any such people, there usually are carts that the airport can provide to help you move your luggage.
The last option is to ship your luggage in a more conventional way. Is there a friend or a hotel that can accept and hold packages at your destination? It may be much less hassle, and especially if you're moving permanently, it may also be much, much less expensive.