Just oil the cogs and brush them with a wire brush. Then clean them off a bit with some mineral spirits or WD40 to get rid of the brown rust sludge and re-oil them along with the chain. You don't need a fancy bike oil for this either. Just some regular engine oil will work if you have some. But if you have to go and buy something anyway then get a quart of Mobil 1 synthetic engine oil and you'll have a lifetime supply of top quality oil for your bicycle chain and many other uses. Get the thicker 20-50 if they have it so it doesn't wash out as easily.

And obviously if you value the investment in your bike you won't leave it outside anymore. It's one thing to ride it in the rain or even leave it locked to a bike rack in the rain. But leaving it out for storage in the rain is unforgivable on a lot of levels.