Grimesquit & Phingspet
Start by talking to Gertrude. She will direct you to some rat catchers in the Varrock Sewers to give you some rat catching tips.
Grimesquit and Phingspet will be right near the entrance in the sewers. They will give you a hard time, but will eventually tell you to catch 8 rats.
After catching 8 rats with your cat, they will tell you that they can't really teach you anything there, and will instead direct you to Jimmy Dazller in Ardougne.
Before telling you anything, he will ask you to help him out. He wants you to get rid of some rodents in a merchant's house, but you can't be seen. When ready, read the instructions that he gave you.
To the northwest, when the two guards are away from each other, run across the bridge. Follow the fence north, staying behind the bushes. When the guard isn't looking, run north behind the other bushes, and look for the trellis on the back side of the manor.
After climbing up the trellis, head into the west bedroom and dispose of the rat. Head to the southeast upstairs room and get rid of the two rats, then go down the ladder and get the one there. Go into the kitchen and get the last two.
Return back up the ladder, and climb down the trellis, and you will be back at Jimmy. Talk Jimmy again and he'll allow you access into the Ardougne Rat Pits for kittens. He will also direct you to Hooknosed Jack in Varrock for grown cats.
Talk to him, but he will be too busy to help you out. He will agree to let you take on one of his jobs of poisoning rat holes in a warehouse to the south. Give him the kwuarm, red spiders' eggs, and vial, and he will give you rat poison.
You'll need to poison some cheese to use as bait. Head south of the nearby bar and enter the building. Climb up the ladder. Use the rat poison on the cheese, and use the cheese on each of the rat holes. Once all the holes have been poisoned, return to Jack.
Pox has been poisoned. Ask if you can help. He will direct you to the apothecary in western Varrock to get help with a cure.
Talk to him about cats. Give him marrentill, a ground unicorn horn, and a bucket of milk. He will give you a vial of cat anti-poison. Return to Jack and give him the anti-poison.
He will now tell you you didn't finish the warehouse job. He says there is one rat left. Return to the warehouse (and make sure you have food for your cat!). Use your cat on the hole in the wall.
Your cat will fight the king rat. WARNING! If you aren't careful, your cat could die and you will lose your cat! To supply food to your cat, use the food on the hole in the wall.
After the rat is dead, return to Jack. He will now allow you into the Varrock Rat Pits for grown cats. He will also direct you to Smokin' Joe in Keldagrim.
Find Smokin' Joe in Keldagrim, near the pickaxe shop. Offer to help him smoke the rats out. Use weeds on your empty pot, then light it on fire. Use the smouldering pot on the nearby rat hole. The rats will run out the otherside, but will go back in when you're done.
Try again, and your cat will talk to you. It will offer to catch the rats on the other side as the are smoked out. Talk to Smokin' Joe again. He will give you access to the Keldagrim Rat Pits for overgrown cats, and direct you to The Face is Port Sarim.
The Face & Felkrash
Talk to The Face. She will direct you to Felkrash below in the pits. Enter the bits and head to the east. Felkrash is in the blue. She will offer to teach you how to make your cat wily... after getting rid of every rat in Port Sarim...... with just one action. Head outside and talk to The Face again, and she will tell you of the snake charmer.
Travel to Pollnivneach, and find the snake charmer just outside the bar to the south. Use a coin on the money pot next to him. Ask about charming animals. He will refuse to tell you how to charm rats, but offer him some money. He'll want more than you have, so tell him forget about it. Choose to walk away slowly. Then choose to stop. You'll pay him 100 coins to learn how.
He'll give a music sheet and a flute. Return to Port Sarim, and standing near the entrance to the rat pits, play the snake charm. You can only play 1 note per page, so each flute shown on the music sheet is one note. On the first page, select the far left note, then go to page two and play the fourth from the left note, and so on. Don't forget that page 5 needs to be a higher octave!
When ready, hit Play! Rats will appear out of no where, and you will lead them into the water. Talk to Felkrash, and she will tell you she can only turn overgrown cats into wily cats.