Check with your close friends, acquaintances and associates to provide you with vital information on professional movers. Weston Moving & Storage (phone:- 954-349-1916) has three offices in Florida and they serve internationally, they have a good name as a professional mover.
There are international moving companies that can help you, that even if they are not based in your state, they can still provide the service you need. One of the best known international moving companies is globalrelocations.com. It can help you pack and ship your belongings internationally. Their handling capacity is much bigger than other moving companies. They also offer customize packages for your travel depending on your needs.
In Canada, it is tough to find one, as you call the best moving company. There is a list of moving companies that the Govt. has published where in you can find the trusted ones. I actually checked with my friends and also checked on their reviews before renting the moving service. Major Move (Ontario) www.majormove.com has another exception of having good packing facilities which you can rent or buy, according to the need and it was cool. They have these wardrobe packing that my wife liked a lot.
You could start with a local search, either online or using a phone book. A-1 Auto Transport is one long distance mover that has a strong presence in the area and they also haul cars which can be helpful if you're looking for that. If you only need a local mover then there's no doubt several cheap options available to you.