Using staves does not increase your skills, nor do staves benefit from higher skill levels, with the exception of lowering the cost per use.
However, the higher the user's skill level in the relevant magic school, the lower the charge cost to use the staff. Anyone can use a staff, regardless of magic skill, and using a staff does not use any of the caster's own Magicka instead, it depletes the staff's charge. Staves are one-handed weapons that are essentially rechargeable scrolls.
