How to catch bugs and critters

You have to walk to them slowly, don't run or they fly away. You don't need any tool to catch them.

Edit! (Thanks to The Undisputed Hero) : You can catch them by running, but it's a timing question. Also, bugs that flies can but caught by jumping.

Getting a pet

You have to wait for the summer on your first year.

Training a pet

The best way I found was to give them treats. Then you play a minigame.

Getting a horse

You get one free in the middle of summer

A couple of answers to some questions thanks to user cerity

Q: Can you move animals from one barn to another?
A: YES! You can push big animals or pick up chickens and place them in the other building.

Q: How do you train your horse? Do you just run around?
A: You have to interact with the board right outside the horse track. You can train 3x per day before 10pm. The horse treats are alternative food and have nothing to do with training unlike pets.

Q: Why have my trees stopped gaining quality despite fertilizing everyday?
A: I just did an experiment, and it seems they only increase quality during the growing phase. If they are mature, the quality can't go up.

Q: My 5-star dish didn't win the cooking contest, what gives?
A: Complexity of the dish really matters! Dishes that require ingredients with multiple steps of processing are key, which require PLANNING! (cheese, butter > milk, for example)

Tea

You need to unlock the yellow windmill. (Bazaar rank 4)

Honey

You need to unlock Arata's stall in the bazaar.