Ok, I think I'm getting somewhere- the suggestions about lowering the shifter pedal seem to have made a huge difference (I had lowered it some, but apparently not enough). After lowering it more, I found it was much less likely to miss the 1-2 shift. I think maybe that combined with 12 years of over-reliance on my last bike's heel-toe shifter may explain a lot of what was going on. The rest was probably just panicked overreaction on my part, heh.
I'll probably lower the shifter a bit more after work today, and see if it gets any easier.
I'm still going to implement a lot of the other good advice I got here, though. I may buy the springs and keep them until the winter, when I'll have it apart for paint anyway. And I'm definitely changing the oil this weekend. A cursory glance around seems to suggest that Mobil 1 is the overall favorite, so I'll give that a try.
This community is great, I really appreciate how all of you jumped in to help. Sorry for overreacting, I've just wanted this bike for a very long time.