Something like those pink oysters they sell on ebay might work out. They are some tropical species that grow in room temperatures just fine 
But in general growing mushrooms isn’t easy, especially if you want them to produce fruiting bodies (those visible edible parts). For that you need correct temperatures (I remember reading that some species need correct diurnal temperature fluctuation to start growing their fruiting bodies, not just constant cold) and humidity. These growing conditions might not be available near your house for the local mushroom species you might want to grow, unfortunately. At least here in the north, they (almost) all want to grow only during springs and autumns 
But if you think you might have the right amount of shade and enough humidity at your backyard, then you should give it a go, though. Just don’t keep your expectations too high for the first few trials and errors and don’t get too discouraged by what I said on former paragraph 

Here’s few tips I learned when I tried to grow Kuehneromyces mutabilis few years ago. This species grows in dead wood logs:
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You need to keep everything absolutely sterile, meaning growing media, cutting tools, table surface etc.
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You need to collect the spores from mature and healthy looking caps. For some species you might want to take the actual rhizome itself with sterile equipment from the inside parts of the mushroom.
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You might want to try growing multiple tissue cultures first and inoculate (transferring of this grown tissue stuff) on the wood material later. This way you’ll avoid introducing the same amount of competitor spores from air as the wanted spores. The tissue would have much more mushroom cells so it gets a nice headstart to outcompete the other molds and mushrooms.
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The first mushroom that wins the chunk of wood for themselves gets so good competition advantage that no other mushrooms can get in and start growing in that same part of wood later, until that mushroom has eaten pretty much everything it cans and starts dying off. For this reason the wood logs needs to be pressure cooked too, to kill off those species that might have already started growing their rhizomes inside the wood matter, they can’t be seen with naked eye.
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Bonus: Don’t buy the ingredients you need for tissue cultures from the same pharmacy shop and don’t tell you’re going to grow mushrooms. They’ll just think you’re going to grow some illegal stuff because usually people don’t bother growing mushrooms themselves and instead just go pick them from somewhere 

Good luck if you’re ever going to give it a shot 