i know there's an obvious solution to this, but i don't usually work on bikes with gears and i can't figure it out. i'm rebuilding an old 10 speed, with downtube shifters, and when i reinstalled the derailleurs + cable + housing, it doesn't work right. it'll shift when i push down on the shifter, but as soon as i let go it springs back. i have it so the shifters would move towards the back of the bike when you wanted to downshift...is that wrong? i don't see how just flipping the shifters would fix the problem though....
'''96 Litespeed Catalyst, '05 Litespeed Firenze, '06 Litespeed Tuscany, '12 Surly Pacer, All are 3x8,9 or 10. It is hilly around here!
Downtube shift levers are held to the mounting bosses with either a large slotted screw or D-rings. They have to be tight enough to hold the lever in position against the derailleur's springs. Snug whichever type you have until the lever holds it's position but you can still move it with reasonable force when you want to shift.
The rear shifter shift lever should move as you describe, i.e. move the lever rearward to shift to a larger cog. The front shifter moves the opposite way; i.e. moving it forward shifts to a smaller chainring.