Haha, Im far from the master
If you think you have a suitable place in your yard, you could simply remove the section of earth you find them in, and place it there. Understand though, that the mycelium could be covering a huge area around the actual fruiting bodies that you find, and only in that one small area does nature balance perfectly for the fruit to grow. Lighting, temperature, humidity ect. all have to be perfect, and each species is different. If youve been hunting them awhile though, you may have a good idea of what sort of conditions you will need to provide.
You could also just retrieve a sample of the buried mycelium, and add it to a a medium youve prepared, or purchased, and pick a seemingly suitable place in the yard to keep it.
Or...
The most sure fire way to do it, would be to prepare a sterilized medium in jars, add the mycelium in, let it grow, dump out the jar, and let it fruit in a sterilized area indoors.
There are alot of issues with trying to propagate fungi though. Like I said, the conditions have to be just right, but youd be surprised how many people have success with just moving the patch of earth into there yard, giving it minor attention, and having them come up right there next season. The sterile indoor method is much faster, and would likely get better results, but if things arent completely sterile, it all goes to pot quite quickly. So its just a bit of a pain.
The general rule is to leave the mycelium as youve found it, that way it continues to thrive, and more fruit comes out next year. Taking a small sample should not hurt though, if you are seriously intent on growing them up. You really dont even need mycelium, any chunk of the fruit should do, just might take longer.
These people seem to know something about the specific fruit youre interested in:
http://thegreatmorel.com/growingtips.html
You can google around for more info on mediums and setups ect.
There are people that grow greenhouses full of rare fungi, and sell them off to restaurants around the world for a mint.
You can buy a spore sample of just about any species you like online.
A very interesting hobby to get into all around, but very demanding...