Hi Jan,
My ability with a jeweller's or piercing saw is possibly due to the fact I worked in the jewellery trade. The following might help you but using the saw a lot is possibly the answer.
First as I expect you know the blade cuts on the back or down stroke.
The saw blades must be of good quality and have an even set if they don't they will cut curves when held straight. The backs of the blades should be curved so that they don't jam easily and are more maneuverable. The ones I use are Swiss and cost about £20 a gross, oddly for a metric country they are wired in dozens twelve to a packet. They do split the packets but cost more. They can be bought from Horological suppliers in the UK I use
www.cousinsuk.com who sell Shark and Bergeon I use the Shark they are a little cheaper.
Having got a decent quality blade there are about 14 different tooth pitches, which is important when cutting thin material, a very minimum of two teeth should be in contact with the metal when cutting. I use 0/3 which is fine enough to cut K&S tubing without jumping and 3 for thicker general purpose work. I have a 3" frame for each size as it saves time and after putting a used blade back into the frame they usually break
The saw frame. There are several sizes generally increasing in size in increments of 1" from 3" upwards. When making long cuts start with the smaller frame and go onto the large when necessary. I use a 3",4" and a 6". The larger frames are heavier and have more spring when being used.
Fitting the blade, there is usually a projection at the top of the frame in front of the upper wing nut this is to put into a hole drilled in the front of a bench and makes it easier to put tension on the blade while pushing on the handle and tightening the wing nut. Tension should be as tight as possible the less spring the better.
As often the wing nuts get in the way when using the saw I replace them with nuts that need Allen keys.
Using the saw. When cutting it is a good idea to be sitting down with the work piece at shoulder height. It is a also good idea to clamp the work piece to a peg projecting from the bench. When cutting straight lines the saw is slightly inclined forward and only used vertically when cutting curves or changing direction. When cutting thin sheet clamp or double side tape the work piece to some fairly thin plywood and cutting through both helps to stop the work piece vibrating. When cutting a lot to save having to stop to clear the debray so the guide line can be seen a fish tank air pump connected to a silicon pipe with a weight at its end can be used to do this job. The weighted end of the pipe being directed along the line of cut.
If anyone is still reading this missive what was said at the beginning about using the saw a lot is very important. If anything isn't clear please ask.
Regards Tony.