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Assuming recycling is possible, is it more costly to the environment to throw waste in the garbage, or to recycle it? Does the answer depend on what kind of recyclable waste it is?

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Yes, it depends. It depends not only on the type of waste, but also on the methods for disposal.

Example:
Aluminum is the perfect thing to recylce. Takes no more engery to recycle, and no loss in product quality, than to mine it.

However, plastic degrades as you recycle it. And, it takes more energy to recycle than to make fresh from petroleum. Particularly while petroleum is relatively inexpensive. Someday this will change.

But, there are different ways to dispose of things, too. The current landfill method does not allow things to biodegrade, thus, it uses space/land that cannot be reused. And it also tends to pollute such space/land/air/water.

Composting is more efficient. The trouble is regulating what goes into the compost pile since people tend to throw away hazardous materials (batteries, diapers, oil) that polute and destroy composts.

Edit:
Consider this hypothetical. Would you rather have 1000 pounds of plastic crunched up and burried in a hole or turned into a park bench? Is the park bench recycling or creative waste disposal?