A saw that will not cut straight is nearly always being pushed. Hand saws cut on their own weight and the stroke should feel almost effortless; force introduces flex, and a flexing blade follows whatever path the flex suggests rather than the line you marked.
Rip and crosscut teeth
Rip teeth are filed square across, like a row of tiny chisels, and they cut along the grain by paring it away. Crosscut teeth are filed at an angle to a point, and they sever the fibres across the grain like knives. Using the wrong one is not dangerous, merely slow and rough: a crosscut saw ripping a board will take three times as long and clog.
Tooth count tells you the rest. Fewer, larger teeth cut fast and rough, more teeth cut slowly and cleanly. Coarse for rough carpentry, fine for anything that will be seen.
Starting the cut
Most errors happen in the first inch. Start with the saw almost flat, pulling backwards a few times to establish a kerf, with your thumb knuckle as a guide against the blade. Once the kerf is established the groove keeps the saw honest. Begin by pushing hard and the saw will jump and tear the edge of the work.
Why it wanders
A saw that pulls consistently to one side usually has uneven set, meaning the teeth are bent further out on one side than the other. It can be corrected but it wants doing by someone who sharpens saws, and on a cheap hardpoint saw it is not worth it.
If the cut drifts gradually, check your stance rather than the saw. Your eye, shoulder and the blade want to be in one plane; standing to one side of the cut makes drifting almost inevitable, because you are sighting the line at an angle.
Hardpoint or sharpenable
Modern hardpoint saws have induction-hardened teeth that stay sharp considerably longer and cannot be sharpened at all. When they dull you replace the saw. Traditional saws can be sharpened indefinitely but need doing. Neither is wrong: a hardpoint for site work and rough cutting, a sharpenable saw if you enjoy maintaining tools and want one that improves with use.
Looking after them
The teeth are the only delicate part. Saws stored loose in a drawer or leaning against a wall lose teeth against other tools. A slit of wood over the teeth, or hanging the saw by its handle, is all that is needed. Keep the plate lightly oiled, because rust on a saw plate increases friction noticeably and a stiff cut gets blamed on sharpness.