Im a tree man , I have worked as a sawyer both harvesting that wood and milling.
If possible cut hardwood deciduous trees in the winter when the sap is down, this will save months of dry time.
As far as what to build with, if the wood will be in contact with the ground or buried in the ground use Cedar, Locust, or redwood. Paint the area below soil level and one foot above with the baddest paint you can find, Barn paint or something stronger. You can also go to a local asphalt plant and buy some driveway sealer, dont get the crap you buy at home depot, get the stuff that never comes off anything in gets on. Most plants will call the stuff Tack oil and it will be either RS Rapid set or SS slow set paint the area to be placed in the soil with it.
now as far as the above ground structure goes there is nothing wrong with , Ash (Right now its very avalible due to the Emerald ash bore), Oak, walnut, chestnut, hickory, locust, maple, Cherry, Osage orange(Hedge apple) or pretty much any hard wood deciduous tree.
Don't use Pine, Birch, spruce or other sappy evergreens.
If you mill the wood remember you will need to let the wood dry and will have to maintain the correct stacking procedure turning the wood once a year. You will loose a foot off of each end of the board due to cracking and splitting, so cut the wood long. The wood will need good airflow and protection from the rain so a tarp is the least ideal way though its better then exposed to the rain. A barn is best, its what we use.
You will most likely find that after you mill the wood you could sell one 3rd of it and buy suitable commercial lumber. Its almost a waste to use nice hardwoods for framing unless you are going for large timber framing and desire the look that you get from nice hardwoods, in that case you probably dont wanna build a shack.
If you use the logs you should still allow some dry time, its best to strip the bark from the log as well. This will allow you to properly seal the log using something suitable for deck sealing. You will need a few special tools to do this so look in to them.
Don't use Nails, man if you go through the trouble use mortise and tenon joints. I like to see the darkest wood used for the dowel as it just sets out how amazing the construction is.
I have been part of a few construction projects just like this so if you need some help ill tell ya what I do know.