I try to sow tomato and other above ground crop seeds in the first or second quarter (before the full moon). I also try to sow them during a good sign for such seeds, that is, Scorpio, Cancer or Pisces. These are called the ‘water’ signs and are supposedly more fruitful.
For below ground crops, I try to plant them during the third quarter moon (right after the full moon). I try not to plant anything during the fourth quarter. The best signs for planting root crops are Capricorn, Taurus or Virgo, which are called the ‘earth’ signs.
The theory of planting by the phase of the moon is that during the first two quarters, energy (or gravitational pull of the moon) and water is being pulled up, so that encourages seeds to burst their seed pod and sprout better.
Conversely, during the last two quarters, energy (or moon’s gravity) and water is being pulled down. So, with root crops you want them to start off with them being ‘pulled down’ so that they establish their root type growth. The reason for planting them in an earth sign, is because they do most of their growth underground, or in the earth. Supposedly (not quite sure if I’m explaining this right), you don’t want to plant taters during a water sign or they may have a tendency to rot. My grandmother claimed that if you planted something like carrots or radishes during the waxing moon, you’d get more above ground green growth, and not so much root growth.
Now, to directly answer your question about setting out seedlings. Since the tomato seed has sprouted and grown into a plant, you’d want to help it establish a good root system when it is set in the garden. So, because you want good root growth, transplanting in the third or fourth quarter would be best. The better sign would be an earth sign, but a water sign would be okay. I’m not as picky when transplanting, but it prob would be best done in the 3rd and 4th quarters.
There are the ‘fire’ signs, in which you don’t want to plant anything, as they are more barren. During this time, it’s better to cultivate or weed, so as to help prevent weed growth. Also, a good sign to harvest crops (or fruit), as it supposedly helps delay rotting a bit.
There are also ‘air’ signs, which aren’t as barren, but aren’t ideal for sowing or transplanting. Flowers tend to do better sown during these signs, in the first two quarters.
Ok, now you know my methods (or madness). Just call me hippie earth boy. Guess I got it from my grandma. I don’t follow it to a T, but do try to plant and transplant in the correct half of the moon.
Here is a site that explains it better than I can, there are others if you want to look them up.
You can get almanacs that show what sign the moon is in. You can also get some calendars that will show the signs and moon phases.