After a devastating storm, the most important thing is that you and your family are safe. Of course, it would be nice if your home remained safe throughout the ordeal too. Even if the interior of your home was fortunate enough to escape damage, the exterior and your garden are likely to be a different story.
Cleaning up after a storm is an ordeal in itself, and the safest way to prepare any damaged items for collection in a manner that makes them easy to recycle is to arrange them into piles, before contacting a rubbish removal services company.
Fallen branches are easy enough to handle, but when an entire tree has been uprooted, it's a different matter entirely. Consider renting or borrowing a chainsaw (if you don't own one already) so you can safely cut the tree into manageable portions. These portions can then be dragged onto your front lawn and placed in a pile. Wooden fencing should also go in this pile. If possible, strip the foliage away from the branches and place this in a second pile for garden waste.
The garden waste pile can also contain any damaged, smaller plants. Since this pile will exclusively be comprised of garden waste, it can be composted. If you compost at home (and if your compost bin is still intact after the storm), consider filling it up with material from this pile. Any excess should be left for collection.
You might need to have another pile for mechanical debris. Ideally, you would not need this pile at all, but there are a few mechanical devices that might be unsalvageable after a storm. For example, the exterior components of your split system air conditioner might need to be removed and disposed of, or the pump motor for your swimming pool. Only dismantle and pile these items if you know precisely how to do so, and only once the electricity has been turned off. If there's any uncertainty on your part, it's best to leave it to the professionals. There is also the possibility that the effect is purely cosmetic, and the item's components are still perfectly functional under their casing.
Metal debris will need their own pile, and this will generally include any parts of the exterior of your home that have been detached from your house. Damaged metal garden furniture and fencing can also go here. Any textiles (such as the cushioning on your garden furniture) should be removed and given their own, smaller pile.
Cleaning up after a storm isn't going to be fun, but the most efficient way to do it is to simply make a few piles.