The black wheel on the furnace does not always spin on it's own, and then the heat does not come on when we turn up the thermostat. If we start to spin it manually, it will kick on and spin on it's own, and then the heat kicks on. Could you tell me what's wrong and if we can fix it on our own?

By black wheel do you mean the inducer blower? If so it must kick on before a gas valve will open.

Yes, I think it is the inducer motor. The problem is it won't kick on by itself. I was able to get it to kick on by physically turning it once myself, and then it would start turning faster. That happened a few times last year and this year (and otherwise ran normally, turning on by itself). It happened again tonight, but now I can't get it to start up by turning the wheel. The wheel isn't kicking in and turning by itself even if I help it get started.

An inducer motor isn't something you repair, you replace it. Can you post a picture of it?