Here is a quick list of all the current mounts in Terraria and how to get them

1. Bunny - Summoning item: White Carrot - how to get completing a set number of quests for the Fisherman NPC

2. Slime - Slimey Saddle - a rare drop from the King Slime mini boss

3. Minecart - Tracks

4. Turtle - Hardy Saddle - Found in gold crates

5. Giant Bee - Honey Goggles - A rare drop from the Queen Bee boss

Mounts that can only be attained in Hardmode

Pigron - Scaly Truffle - Fishing

Reindeer - Reindeer Bells - A chance to drop during the Frost Moon Event