I did the oiling at my parents' house, since I don't have much room to spread out in my apartment. (Especially outside. And my parents don't have neighbors smoking six inches away from their patio.)
Here's an unfinished piece of the wheel. If I remember correctly, the wood is New Zealand Silver Beech.
This is what the Tung Oil looks like up close. It has a smell, but it's not a particularly "chemical" smell. I couldn't find anything online or on the bottle recommending ventilation. (Except when mixing the oil with turpentine.) Still, I did my work on the back porch with newspapers spread out everywhere.
I wiped on the oil with one of Dad's old t-shirt as Douglas suggested.
I took this shot to show the difference between the oiled and un-oiled wood. (Is un-oiled the right word? Maybe non-oiled?) I think I used less oil as I went along. This first piece I did is a little on the dark side. I don't know if that's because of the amount of oil I used, or the variations in the wood itself.
Here's a look at that same piece after applying some more oil.
The piece of sandpaper on the right actually came with the wheel, in the folder that came with it. I wasn't quite sure what to do with it. Was I supposed to sand the whole thing? I ended up only using when I came across parts that were a bit rough, which wasn't very often. (Mom says I can sand after oiling, but I'm going to try to avoid that.)
I took this picture right before dinner, to show how far I had gotten.
After dinner, I managed to finish all of the first coat in enough time to shower before my sleep study. Looking at this picture makes me think the variations in color come from the wood, not the way I applied the oil. Some of the lightly oiled parts look pretty dark here.
I didn't oil the grooves in the bobbin or the wheel. Part of me wonders if I should oil the groove of the wheel with one light coat, but everyone else has said not to. (I think oiling the grooves of the bobbins would be next to impossible.)
Also, should I put on only one or two coats on the bobbins (instead of my planned three or four) since they'll be having a lot of direct contact with the yarn?
If you have advice, let me know.