I can try to help you. This game can be difficult to learn at first, but it is worth it once you get the hang of it!
You do not need to make a stockpile for everything in order to use it. Stockpiles are simply for convenience, to move things where you want them. The only thing that really needs stockpiles is food and drink, because it will rot if it is not in a stockpile, and refuse (bones and corpses and shells), because they will rot away eventually and if they are near the dwarves when they do, the dwarves will become unhappy from the smell (which appears as purple miasma). For wood, it is okay to leave it where it is, unless you want to have it all nearby.
As I don't know what is causing your problem, I will go through things step by step, and you can let me know if there is anything you don't understand. To cut down a tree, you press [d] to open the designate menu, then [t] to select cut down trees, then find a tree and click on it if there is only one tree you wish to cut down, or, if you want to clear an entire area of trees, you can move the cursor with the arrow keys to one corner of the area and press [enter], then move the cursor to the opposite corner and press [enter] again. This will designate every tree in that box to be cut down. Designated trees (and plants, and rock) will have a brown box over them.
When the woodcutter (who must have the woodcutting option turned on in their job menu: press [v] to view the dwarf, then [p] then [l] to view their labors, then use the - and + keys to move through the list and hit [enter] to select the job) is ready to cut down a tree, the tree will start flashing. Once it is cut down, it will be replaced with a brown _ which is a log.
To put these logs in a stockpile, press [p] then [w] to select wood, then select an area for the stockpile the same way you designated trees. Dwarves need the wood hauling labor to move the wood to the stockpiles. This can be activated in the labor menu just like woodcutting, and is on by default when you start out. If you wait a while and none of the dwarves start moving it around, make sure you have at least one dwarf idle - the idle display is in the lower-right-hand corner of the screen. If there are idle dwarves, make sure the game is not paused - the game is paused and unpaused with the [space] button.
If my descriptions aren't clear enough, you should have a look at the wiki if you haven't already. The Your First Fortress page has a good tutorial for starting out. This might be easier for you to follow than my long-winded descriptions.