A couple of tips- when you buy yogurt for a starter, you can spoon it into doses and freeze them, dropping one per batch into the warm (not hot!) yogurt. After it thaws out, you stir it in and wait. You can also use the whey for starting the next batch, just freeze it in the mean time.
I drain my yogurt with a clean, sterilized (ie boiled) pillowcase. I turn it so the seam is outside, spoon the yogurt in, hang it off something over a pot, and wait 20-30 mins.
Yogurt 'works' fastest around 110 F, but it will still go more slowly at lower temps. If I'm going to be draining it in 12 h or so, I use an electric hot pad to keep it warm. If I need it to go more slowly, I just throw a blanket over it instead. As time passes, it gets thicker and more sour, so the ideal time depends on taste. There are different 'breeds' of yogurt, which you can try out.
If you're using a thermometer, check it against boiling water. For the mercury/alcohol ones this isn't necessary, but I've found cooking thermometers (thermocouple and bimetallic strip type) that were off by ~20 F, which can mean boiling the milk by accident. As an engineer, the complete lack of calibration on any cooking equipment (temperature, volume, weight) makes me irritated.
As you use each generation of yogurt to grow the next, it will vary in taste over time. Probiotic yogurts will probably not breed true (too many cultures competing), but I've been using Balkan and it works well.