I'm gonna jump on the bandwagon....
Small lathes don't need coolant and don't want coolant. Any time you slow down over heat, it'll get buried and lost in the much bigger "slow downs" you get on a small machine. Coolant works best when it is used daily and serviced (replaced) regularly.
For oil- A sulphurized, or chlorinated (or both) cutting oil works fine for most anything ferrous. I've gone with a sulfrized only cutting oil and/or Marvel Mystery oil (chlorinated but not sulfurized) for steel, and kerosene for aluminum. So I ended up with two cans The Mystery Oil lives by the drill press, it's good there. Cutting oil and kerosene live by the lathe. Aluminum likes kerosene, which WD40 is a close cousin to, and that works well also. I don't go that route just because the smell permiates the whole house so easily, I don't wanna risk it.
Any cutting oil is best applied sparingly with a brush. On a lathe the size of yours, it's not even practicable to pump enough oil to use it directly as a coolant.
Personally, I use oil occasionally. The last couple of light "final passes" setting up to hit a dimension with out (buggering) around trying to sneak up on it for example. When consistancy and surface finish count, both for measuring and for the final product.
Also, for quick little things where I really can't justify making a mess- I keep a block of paraffin wax (gulf wax, canning wax from walmart) and if a part or a tool is slightly warm, it melts in good, lubricates "not great but kinda OK, which is pretty often all you need on a small lathe, it barely smokes and doesn't stink at all. GREAT for when you can't open a door or window for as long as you probably oughtta. Pretty great for woodworking tools too, hand saws, table saw miter slots, hand plane bases, that sort of thing. Used sparingly (or if not, at least respectfully) it doesn't bugger up finishes. Good stuff to have.
My advice is to find a useful can, jar, jug, container to keep your choice of lubricants in, a cheap pack of disposable acid brushes, and give "something a try. It won't take you long to settle in on something that works for what you do.