Even on backyard sawmills with 19" wheels and 1 1/4" wide blades, 7/8" pitch, the blades will wander.
In this case, 99% of the time it is due to the blade getting dull. As long as the blades are kept sharp, with SUFFICIENT SET, there's rarely any trouble.
You are actually starting out with a bandsaw that will have difficulty due to its' small size. Bigger is almost always better for resawing.
Look for a blade with 4 TPI or less if they make such a thing for a 12" bandsaw. Feel the set of the teeth and get one with as much set as possible (within reason) and you will find that it cuts better. Wider is better as well.
You can buy sharpeners and setters, but they are atrociously expensive. I think it would be possible to make your own if you are mechanically inclined. The setter is quite simple, the sharpener is another story.
OK, fess up...which one of you clowns stole my sig? It was right here a second ago.