Bonsly thing:
first of, you need to have seen it on the S.S Libra, after that, it'll appear in the pokespots at random.
if you do spot it, you must move verrrry slowly... just slightly touch the analog stick to make your character move slowly... otherwise Bonsly will be spooked and run away.
once you've approached Bonsly, you'll automatically end up in ONBS and you'll get the Bonsly card for Battle Bingo.
Lucky Egg:
first of, you need to beat the game to get access to Orre Colosseum, you only need to beat the first round. However, this is quite difficult and you'll need perfectly EV trained pokemon to win the round.
once you do that, you'll be able to do the Lucky Egg quest.