What actions do the animals take immediately following the humans’ departure, and why do they do these things?

I assume you are referring to Chapter Two? In Chapter Two, we learn that immediately after running the humans off the farm, the animals immediately destroy anything symbolic of their former bondage....

The harness-room at the end of the stables was broken open; the bits, the nose-rings, the dog-chains, the cruel knives with which Mr. Jones had been used to castrate the pigs and lambs, were all flung down the well. The reins, the halters, the blinkers, the degrading nosebags, were thrown on to the rubbish fire which was burning in the yard. So were the whips. All the animals capered with joy when they saw the whips going up in flames. Snowball also threw on to the fire the ribbons with which the horses' manes and tails had usually been decorated on market days.

After destroying the implements of the bondage, the animals went to the store-shed and ate their fill before settling down to the best night's sleep they'd ever had.

Napoleon then led them back to the store-shed and served out a double ration of corn to everybody, with two biscuits for each dog. Then they sang 'Beasts of England' from end to end seven times running, and after that they settled down for the night and slept as they had never slept before.