I have never heard of anyone transplanting mushrooms. You could try collecting a ripe spore-bearing mushroom and scatter the spores in the right sort of habitat on your property. However, if you have the right habitat, I bet the spores would get there on their own. So, creating the right habitat is key. Each species of mushroom has specific requirements for the places it can grow. Some like rotting wood above the ground, many more seem to grow in organic-rich soils. You could enriching the soil with lots of organic matter and dead wood and see what mushrooms appear. I think it can take many years for a mushroom to appear - fungi can be very slow growers. On the other hand, the types of mushrooms grown commercially grow pretty fast, I think.

Keep in mind that I have never tried to grow mushrooms so I am just guessing here.

Mushrooms are simply the fruit of a large underground mycelium. Transplanting them would be like thinking of transplanting an apple tree by planting an apple.

Gathering mushrooms and scattering them over a suitable area might yield similar mushrooms in 12 to 24 months. The spores from the mushrooms might take hold and start a new mycelium. It will take one, two or more years for the mycelium to grow large enough to fruit. Bear in mind that mushrooms are often very particular about where they grow; for best results you should identify the mushrooms you want to grow then research the conditions that it likes.

There are commercial companies that sell mushroom spawn which is material that has mycelium already growing. You can then dig this into the areas where you want mushrooms. I have tried the indoor kits from fungi perfecti and they did well. They also have outdoor kits:

http://www.fungi.com/kits/outdoor.html

Below is a link to a Canadian company.

Here is a link that might be useful: Mushroom Kits

Since mushrooms grow from the mycelium, one should be able to transplant that and have muchrooms growing??

I think you'd have a hard time finding a small enough mycelium to transplant. Have you ever seen mushrooms pop up from the lawn in a circle (aka 'Fairy Circle')? That shows you the diameter of that particular mycelium. I don't know if a piece of mycelium would work or not.