Bicycle Mechanics - Bike changes gear on the uphill

I have an old Schwinn 10-speed I just picked up. I took it out on some hills today and noticed that it changes gears back and forth when I'm really pedalling hard to get up hills. It's quite obnoxious, but only does it when pedalling very hard. Once I've cleared the hill, it goes back to acting just fine - no rattle or anything to indicate it's not fully in gear. Is the extra torque from the hard pedalling causing it to slip on the gears? What can I do to remedy this?

does it actually change back and forth in gears, or does it "jump," a moment when the your change no longer engages the gears and your pedals sort of jerk? if it's the latter, measure the chain across 24 pins, it should measure out to 12 inches. if it's longer than 12 1/8 you need a whole new drivetrain. less than 12 1/16 you might get away with just a chain. also shift the rear into the smallest cog and rotate the crank backwards, keeping your eye on the chain at the cog/rear derailleur. you might spot a sticky link, which can also cause a skipping problem, although usually regardless of load.

if the bike progressively shifts into a harder gear, check the screws that hold the shifter assembly together. if they are loose, the shifters cannot resist the tension that the derailleurs provide.

It does a rough shift upwards, then will do a rough shift back down. It is jerky, but it is like I'm shifting while riding up the hill (like when you shift while putting a lot of pressure on the pedals). I don't know if it does it across all gears, because I'm usually in the same gear region when I'm hitting a hill.

I don't really have the money to put into new parts for this bike - I only paid $10 for it. Is this something I can just learn to live with and not fear an explosion?

I don't really have the money to put into new parts for this bike - I only paid $10 for it. Is this something I can just learn to live with and not fear an explosion?

Try cleaning the chain and relubing it.

Also, I assume it's friction shift so check the tightness of the bolts holding the shifters onto the downtube.

about the only thing that will cause an explosion on a bike is an overinflated or unseated tire. if you want to ride it 'til it drops, ride it 'til it drops.

What about that band called "This bike is a pipe bomb"? :)

haha, i thought of them after i posted. i used to see their stickers on bikes all the time and didn't know what it was about.

Let's suppose I was interested in replacing components on this bike - do modern components fit? I got to thinking that maybe I could slowly relplace everything with some nice components, and then eventually get a new frame, essentially phasing this bike out into something brand new.