It really depends how particular you. If you paint or oil it once assembled you won’t seal the ends and the wood will be more prone to absorbing moisture on the unpainted/unsealed ends. Whether or not this is an issue for you will in-part depend on your location and in-part on your personality.
If you choose to use oil, use 100% Tung oil. Put at least 3 coats on and wait a few hours to a day between coats (whether dependent). A light sand between coats is also recommended. Note that you may need to reapply the oil every few years.
If you choose to paint, use a water based paint that has no fungicide in it. Apply two coats minimum based on the instructions (usually waiting a few hours between coats).
DO NOT oil or paint the inside surfaces.