The picture isn't bad at all. It looks very nice, and these old straight stitchers usually stitch perfectly on most types of fabric. The only thing I would be extra careful with is the top tensioner on the side; the early Whites tend to have a rather fancy mechanism, and I would simply vaccuume around it and not take it appart if it's not strictly necessary.
The throat plate should come off to clean around the feed dog and bobbin area. There's probably oil pionts several places in the body, and if a cover or plate looks like it should come off, it probably does. Look for accessable internal parts; all joints, gears, levers, hand wheel, rods; moving parts where metal touch metal needs oil. There's usually oil points on top of the machine, behind the face plate, and under the base. The presser bar needs oiling, as well as needle bar. Move knobs and levers, turn the hand wheel, behind them there are parts that need oil.
The staining on the body should clean up nicely. I had a cricle finish case I used liquid car polish on, the whitish type, and it was difficult to buff up after it dried. Try it on a spot to check how it turns out, if the finish isn't too crincly it might not be a problem. The glace or sealant version (often sold as a second step after the basic polish) doesn't leave the whitish grit behind in the cricles.
I'm guessing there's a rubber motor pulley running on the inside of the hand wheel some where, it usually needs replacing on these old machines. I couldn't find a free manual online, but there is at least
one pay version available. When there's no manual or tutorial available you just have to use your intuition. I'm sure somone here on QB have a similar model, and there's clever people on the Victorian Sweatshop forum too.