Gold
The Golden Machine takes a minimum of 10 of the same normal pet and makes one gold pet. The listed power increase is about 13%. The diamond cost in the article is 0, so you can convert as many sets as you own.
Locations disagree between two pages, so check both. The machine list puts a Gold Pets Machine in Area 10, the Mine. The Golden Machine article unlocks it at Area 31, Desert Pyramids, and also lists copies at Area 100, Void Spawn, and the Supercomputer.
Shiny gold pets have to be made from shiny pets. Mixing a shiny into a pile of normal pets, which Pet Simulator X allowed, is described as removed.
Rainbow
The Rainbow Machine takes a minimum of 10 of the same gold pet. That is 100 normal pets if you convert every step instead of hatching golds. The listed power increase is about 27%. The tooltip that says rainbows are 10× better is called out as wrong. The machine is free. It is in Area 31, Area 100, Void Spawn, and the Supercomputer.
Shiny
Shiny is a separate variant from gold and rainbow. This page does not have a hatch rate for it. For the Garg Machine, a shiny pet adds 3 points on top of the pet's strength points. Blood Moon rewards use names such as Rainbow Shiny and Golden Shiny, so those variants can stack on limited pets.
Fuse
The Fuse Machine is in Area 28, Shanty Town. It combines 3 to 100 pets into one stronger pet. Secret pets and Exclusive pets cannot be fused. Limited pets cannot be a fuse result. Gold and rainbow results are possible. Shiny results are not. Fusing Mastery can later guarantee gold or rainbow results, add a small chance at two pets, and lower the minimum from 3 pets to 2.
Garg Machine
The Garg Machine, also called the Huge-A-Tron, is in Area 3, the Castle, and needs the Castle Key from Rebirth 1. You deposit Exclusive pets from the Exclusive Shop or Exclusive eggs. Achievement pets, Huges, and Titanics do not go in.
Points come from the pet's strength percent: 85% gives 1, 90% gives 2, and 95% gives 3. Gold adds 1 point, rainbow adds 2, and shiny adds 3. You need at least 50 points. Thresholds are 50, 75, and 100. The result is a Gargantuan Machine Egg that hatches a Huge, with a chance at a Titanic and a Gargantuan. The cost table shows 800, 300, and free at those three thresholds. The excerpt does not name the currency, so it is not assumed here.