As PatrikLundell says, if you need to move items out of a stockpile into another stockpile, out of the temporary initial general storage into a dedicated stockpile for example, use a link. Another option would be to disallow the item type in the old stockpile after building the new stockpile.
To add to what YetAnotherLurker has said - If the item type has configurable material and quality properties, make sure you set those up as well. If you have a stockpile beds, but haven't specified the allowed bed qualities, no beds will be allowed at all.
Yet another thing to watch out for, and this one has caught me off guard and caused quite a bit of frustration on multiple occasions - Some item types have an option to allow usable and unusable items. Off the top of my head, this will apply to weapons and armor. Make sure that this setting is set where appropriate.
In the case of a stockpile that stores liquids, such as drinks, make sure that allow barrels in the first stockpile settings screen and not in the custom stockpile settings screen.
Past these issues, we would probably need to see a save or some screenshots to diagnose the exact problem.
Tangentially related -
Wheelbarrows should only be used on stone and wood stockpiles. They do not do exactly what you probably think they do, and you should check the wiki for details on how to use them correctly.
Some items can be stored in bins or barrels, and a lot of items cannot. You don't want to enable bins and barrels on stockpiles that don't actually use them. Doing so will make the dwarves put an empty bin/barrel in the stockpile, wasting both the bin/barrel and a stockpile square. It will probably take you some trial and error to figure out what is what.