You CAN bring multiple pets and different varieties of pets such as kneazles, pygmy puffs and nifflers to Hogwarts in your second year at Hogwarts. A problematic pet like a dragon or acromantula won't be acceptable, but as long as you can take responsibility for your pet's well  being and the well being of others while your pet is around and for your pet not harming anyone or anything at Hogwarts, your pet will likely be acceptable.

You CAN bring your own broomstick to Hogwarts and fly without supervision at Hogwarts in your second year at Hogwarts. You CAN also play Quidditch at Hogwarts and join the Hogwarts Quidditch teams in your second year at Hogwarts. However, you will be supervised whenever you fly if you have not yet obtained passing grades in Flying classes for first years at Hogwarts.

You CAN apply to become a Professor Assistant for the Professors of first year subjects at Hogwarts.

You CANNOT visit Hogsmeade yet.

You CANNOT apparate or obtain an apparition license yet if you are not seventeen years old.

You still CANNOT practise magic in front of Muggles and still CANNOT perform underage magic outside Hogwarts.

You CAN NEVER bully or harm another creature in or outside Hogwarts, no matter how many years old you are or in which year at Hogwarts you are.