There are 29 different slimes in total, however only 24 of them need to be killed. The other five slimes are from Christmas and Halloween events. The slimes can be found as follows.

Pre-Hardmode slimes
Baby Slime: A few Baby Slimes will spawn after killing Mother Slime.
Black Slime: Spawn in the Cavern layer, and are hard to see.
Blue Slime: Found in Forest and Underground, or summoned with a slime statue.
Dungeon Slime: Found in the Dungeon and have a Golden Key inside.
Green Slime: Found in the Forest during the day.
Ice Slime: Common enemy in a Snow Biome and also appear in the Underground Snow Biome.
Jungle Slime: Common enemy in the Jungle biome.
King Slime: Summoned with Slime Crown as well as by very rare random spawns when a player is in the two outer thirds of the map (far left or far right), the depth is above zero, and there is standard, green grass as a potential spawn point. Note: Goblin Scouts will spawn in the same locations as King Slime, indicating the proper location has been found.
Lava Slime: Spawn only in the Underworld. They have fiery glow and emit some light.
Mother Slime: Second largest ordinary slimes. They spawn in Cavern layers.
Pinky: Very rare slime that can appear in the Forest biome. It is the smallest of the slimes.
Purple Slime: Spawns on surface uncommonly in the outer thirds of the world.
Red Slime: Found spawning in the Underground.
Spiked Ice Slime: Commonly found in the Underground Snow Biome.
Spiked Jungle Slime: Can be found in both Jungle and Underground Jungle Biomes.
Umbrella Slime: Found during rain on the surface. They are larger than Blue and Green Slimes and hold an umbrella.
Yellow Slime: Found in the Underground layer.

Hardmode slimes
Corrupt Slime: Can spawn in both The Corruption and Underground Corruption Biomes. Similar to Mother Slimes but stronger. It is a Corruption counterpart to a Crimslime.
Crimslime: Found in the Crimson biome in different sizes. It is a Crimson counterpart to a Corrupt Slime.
Illuminant Slime: Spawn in the Underground Hallow Biome, replacing normal cave slimes. They emit a small amount of light.
Rainbow Slime: Rare spawn near The Hallow during rain or blizzard. They will spawn outside of The Hallow Biome while a player is physically inside. It does not spawn directly on the hallowed blocks, but on non-hallow blocks such as grass or snow.
Slimeling: Three Slimelings will spawn after defeating a Corrupt Slime.
Slimer: Spawn in The Corruption and Underground Corruption Biomes. It begins as a flying Slime with wings but will lose them and fight you on ground.
Toxic Sludge: Spawns in the Underground in place of ordinary slimes.

Note: Because the Corrupt Slime is only found in a Corruption Hard mode world and the Crimslime is only found in a Crimson Hard mode world, you cannot earn "Gelatin World Tour" with a single world. To avoid missing any of them, have 17 out of 24 slimes before you trigger Hard mode. Then, collect up to 23 out of 24 of the slimes in your original world before travelling to a second world and immediately starting hard mode at that location for the last slime.