Each phase has a different personality, or a more specifically, a heightened mood, depending on the phase. Each phase lasts around two or three days, usually the day prior, the day of, and the day after the specific phase occurs. Sometimes these phases can begin a few days before and/or last a few days after.
Regular Luna is as is
Blood Moon Luna is more serious and focused on her work. When she’s in her Blood Moon phase it’s probably wise not to joke around with her.
Blue Moon Luna is more calm and mellow. She’s heavily focused on work, just as Blood Moon, just not as stern.
Full Moon Luna is stressed and a little paranoid. Her work falters and she gets a bit too worried about things out of her control, convinced everything will go wrong for some reason or another.
Harvest Moon Luna is overly cheerful and rather motherly. During this phase she insists on providing meals for pretty much everyone in the castle, which is a lot of ponies. Sometimes she’ll even roam the streets of Canterlot with leftovers of the copious amounts of food she prepares and give some of it out to random citizens she runs into.
New Moon Luna is pretty grumpy and very irritable. This phase is essentially the embodiment of waking up on the wrong side of the bed. The first new moon of every year is marked as Luna’s birthday, which she’s never too excited to celebrate and usually tries to hide in her room and sleep to escape the torment of birthday wishes and parties. Celestia and Pinkie Pie never let that happen.