You will not need to buy the particular spell, but I think some items may require spell components (bat wings or certain gems for example) in order for their effect to function. It depends on the item.
Some effects are combat only, like on weapons, that work more or less randomly when the weapon is being used in combat. There is almost always some kind of level restriction on weapon effects like this, which means that even though you might be able to use the weapon at level 1, you might not be able to get the weapon to generate its effect until you reach a specific level. Note that some weapons will process an effect at level 1.
Some effects are triggered, which means that you can use them by right-clicking on the item when it's in an inventory slot (not in a bag). Some items with triggered effects have charges which limits the number of times the effect can be used. Items with triggered effects can also have level restrictions. The level at which you can use the item varies depending on the particular item.
There are also worn effects that are basically always active as long as your character is wearing the item in its respective item slot. I don't think these types of effects are level limited.