Learn Spanish, and if you have children, get them learning Spanish as soon as possible too. Moving to a new country, changing schools and making new friends will be hard enough for them as it is.

Choose an area to live in. You may want to move to somewhere where you will be able to speak English if you have to - if that is the case, choose somewhere near the coast, or near a major inland tourist area such as Playa del Inglés or Puerto Rico in the south of the island. It can be more interesting to live somewhere more typically Gran Canaria though - you need to research as much as you can.

If you own your house. Decide whether you want to sell it and live on the capital for a while, or if you want to rent it out while you decide what to do. The latter option is generally the safest, but either way, you should talk to some local estate agents and start to make arrangements.

Plan your move. If you have children, then summer is a good time to move to Gran Canaria as most schools will not accept new pupils mid-year. In fact you may want to move in May/June, so that you have the best chance of securing a place in the school of your choice for the following September. If you are planning to find a long-term rental property, then book 2 or 3 weeks in a hotel or apartment / villa either a month before your proposed moving date or immediately prior to it.

You are unlikely to be able to organise anything any further ahead than that, as properties tend to be rented out quickly on Gran Canaria.

Check whether your bank will allow you to keep your accounts once you live in Gran Canaria. If your bank does not allow it, move your account to one that does.
 
2 months before you go
Start to sort your belongings and sell or give away anything you do not want to keep. This always takes longer than you think it will!

If you are taking your car, get it MOT'd, and check whether your insurance covers driving in Gran Canaria.

Find an international removals company and buy some boxes (it is not worth trying to save money on boxes - you want the double thickness international freight ones).

1 month before you go
If you are British, then get the new E111 card.

Fill in form P85 and take it to your nearest tax office or your employer.

Unless you are moving to Gran Canaria in April, you should have a tax rebate to collect (since you can claim your full year's tax allowance).
 
Go to your bank and fill in a "Not ordinarily resident" form so that you will not pay tax on your savings - make sure you give them your new address.
 
Pack!