SAE 30 is a good choice for oil. Olive and linseed oil are two of the worst choices - both will turn to gum over time: linseed oil is the oil in 'oil paint'; in addition to turning to gum, olive oil will go rancid and start to stink. Stay away from WD-40, 3-in-1 and other 'penetrating' oils.
Get rid of all loose rust before oiling with a selection of small wire brushes, emery cloth and stainless-steel wool. A small wire brush on a dremel (or a larger brush on a bench grinder) can get rid of rust in gear teeth and racks but remove the rack first or there is a good risk of damaging the woodwork.
You need _very little_ oil. There should be little to no trouble with oil seeping into the wood. Apply the oil to a rag or Q-tip and use that to wipe oil on to the gears. Run the gears back and forth to distribute the oil on to the rack.