At its core, it's just a basic bi-monthly schedule. The Old Farmer's Almanac advocates a lot of maintenance during down times and also has a recommendation for cutting your hair according to the moon's sign. There's a split between above ground external fruit-bearing crops and soil/root crops. There are traditional and modern interpretations of moon gardening calendars with some pivotal differences and discrepancies between the same recommendations falling on different dates.
Growth is supposedly more vigorous during the first and second quarter stages when the moon is waxing toward full.
Obviously, this active phase could be taken advantage of in certain ways. Timing periods of increased growth to coincide with the vulnerable seedling stage when young plants are still becoming established. Fertilizing mature plants or transplanting during the moon's third-fourth quarter waning period (in anticipation of the upward trend growth stage). Weeding or tilling near the mounds, any systemic crop treatments like foliar-applied pesticides. These should all be done when growth is in a neutral phase, up until the new moon.
When I plant in the ground, I always try to do it by moonlight. I think it is easier for the plants to be transplanted at night, and it is easier to see what I am doing if there is a decent amount of moonlight.
I typically plant out in mid-May. The full moon in May is on the 18th, so I'll plant within a couple days of that date.
The new moon is waxing. The apogee was on April 1.
So, theoretically, the influence of the moon's gravity is at a minimum.
Plants' fluids are drawn down into the soil. Above ground portion has less water content than usual.
Today, I pruned several of my in-ground plants. Coincidentally, I think it's a good time for it that is good for pruning to encourage new growth. There are also recommended days for pruning to retard growth.
Also, I have a lot of ripe pods. According to the Old Farmer's Almanac, produce harvested during this time is more binnable ( has a longer shelf-life). Harvests during the full moon have more water content and higher weight, but are more prone to rot.