Each type of saw has its own particular set of characteristics.
You should first decide what you are looking for in a saw and then make your decision based on that.
Saws that cut on the push stroke
How to tell
Typically, if the teeth point away from the handle, the saw cuts on the push stroke.
Characteristics
Generally, push stroke saws are designed for cutting through tougher materials. This is because it’s easier to exert pressure on the saw when pushing it rather than pulling it.
Examples of push stroke saws:
Hand saws for wood
Compass saws
Floorboard saws
Hacksaws
Saws that cut on the pull stroke
How to tell
Typically, if the teeth point back towards the handle, the saw cuts on the pull stroke
Characteristics
Generally, pull saws have thinner blades which are designed for making more delicate and precise cuts. As well as this, the motion of pulling the saw towards you rather than pushing it gives the user more control over each stroke of the saw. This makes it easier to cut in a straight line and achieve a neat finish.
Examples of pull stroke saws:
Coping and Fret saws
Japanese/Pull saws
Please note: The blade on a pull stroke saw is generally thinner and more delicate. As a result, greater care should be taken when using the saw to ensure you do not damage the blade.
Saws that cut on the push and pull stroke
How to tell
Typically, if the teeth are not angled back or forward, and instead, point straight down, then the saw cuts on both the push and pull stroke.
Characteristics
Generally, saws that cut on both the push and pull stroke are designed for fast aggressive sawing, and will cut and remove more material with each stroke. As a result, they will be much less likely to produce a neat finish.