Fib, I sugest that you use the left (front) shifter to put the chain on the middle gear (front one by your pedals). Leave it there. Ride for several days while only shifting the right (rear) shifter. Stay on level ground and feel what happens when you shift. Next week, put the chain on the middle gear in back and go for a ride while only shifting the left (front).

If you want to see what is going on, suspend your bike from hooks on the wall or the garage rafters or have someone hold it up while you pedal with your hand and shift. This is very instructive. You can watch what is happening when all that noise is going on and figure out what to do to make it run quietly.

What model bike do you have? Can you post a picture?