You've got two different double edge swords going on there.
The first being that making the first project on the lathe into a "finished" product is really putting the cart before the horse. Lathes (any lathe) works best after you learn it's character. Unlike wood, these see a LOT of cutting forces, so stuff "moves". Tool selection and/or grinding is equally as important as wood turning, but ten thousand times more options (give or take...).
The second one being that the little mini lathes are inherently made "just good enough", and sometimes not even that. They often need a lot of lovin' to get them up to a standard that will allow you (not the lathe) to make decent parts on. (Or even good to excellent parts, I'm not knocking 'em that hard at all). Many owners find that it takes more investment to bring the lathe up to the standards they want than it did to buy it new in the first place.
Should that stop you? No, absolutely not. But you've got to keep both of those things in mind. For a test thing, to see if you're interested, you've got a great thing there. You've just got to have reasonable expectations, otherwise you're gonna spend an hour with it and loose all intrest in metalworking.
Kerosene or WD40 (which is very kerosene like) work well.
Degreaser like simple green/purple power/ one of those things? Those are pretty fine, they're well safetyfied, but you wanna be able to rinse stuff after. By that, I mean a clean damp rag to follow the cleaner, dry, oil.....
Yes. Bare metal wants to be oiled. Some oil is better than no oil, and on a machine like that, where it can be made to work fine, and even if you do get after that and make it "perfect", it was never built to be an heirloom piece, that advice alone might be plenty good enough. If this is an incidental sideline for you that will get used some, or even if it turns out you use it all the time, you could do better. The first is to stay out of anything automotive. That's not your friend here. Tractor stuff too, none of that. An ISO 46 or ISO 68 hydraulic fluid (R&O or AW would be fine) labeled as such, and NOT lableled for tractors, splitters, or any specialty purpose would make a much more appropriate oil for such things. A gallon would last years. And you don't say what woodworking you do, but if it applies, it'd be pretty ideal for metal slides, trunions and mechanisms on wood working machines that specify oiling as well.
It's a used machine, at a price point that makes several of it's individual components worth more than the whole thing. While I do very much believe in and practice that approach to repairing and maintaining almost anything... I'm not spending 30 bucks on brushes for an off shore angle grinder that costs 35 dollars to replace. These are not hard to disassemble. Evaluate anything obvious for feasibility, but plan on a second teardown. Several things need a lot of tearing apart to get to, but the whole teardown is not bad enough to warrant precautionary investments until you know what you have.
I'd bet that if you take that lathe to pieces, get all of the sticky parts unstuck, get all the sliding parts un gunked and un rusted, and get to a point where you can say that this is possible to piece back together, by then you'll have questions...
One thing I might toss out, if you need tool bits to even try this out- These are very small and light for carbide inserts. There's a high level of convenience in them, but they want faster speeds and larger depths of cut... If it's in your wheelhouse at all, especially to get going and feel it out, I'd grab a couple of high speed steel blanks and grind some basic tools. Or if something came with it, run with that. They all can work, just maybe easier coming up to speed with a ground tool that's dead sharp, than an insert tool which won't be.
) but then I'm a rank, wet behind the ears noob, and the feel for appropriate feeds (manual) and speeds hasn't really bedded in, I'd expect.
