Getting a pet in Last Chaos is easy. Each time you kill a monster it drops money, an item or both. Eventually you will come across a Drake Egg or pan flute in a monster drop. But if you don't want to wait for this to happen you can find them for sale from other players in the town centre for reasonable prices. Also if you are a low level character it is a good idea to get a mentor as they will have acquired so many pan flutes and eggs that they will hand them out with ease.
Once you get an egg or a flute it's time to turn it into a pet. In case you haven't figured it out; a Drake egg will give you a dragon and a plan flute will give you a horse. As soon as you receive either of these items a new quest will be added to your inventory (you can check your quests by holding Alt + Q). Go to Raadol town centre in Juno and talk to Lorraine (an NPC) who will ask you to travel to Draton and meet Archaeologist Jajan who will summon a horse or Dragon depending on the item you present him with.
Your pet will gain one experience for every three seconds you are in battle with a monster, beware though as some monsters will also attack your pet. Levelling your pet is going to take a lot of time and patience. The easiest way to train your pet is to remove your character's weapons and attack a monster of the same level. Having no weapons will decrease your attack greatly and draw the battle out immensely; gaining your pet a lot of experience.
Each time your pet gains a level it is awarded with a technical point. You can spend these technical points on skills for your pet; these can also be purchased from the Animal Trainer.
You will only be allowed the option of turning you pet into a mount at level 31. But once you have your pet trained to be a mount by Randol's Animal Trainer it will no longer gain any experience or technical points. For this reason most players choose to continue levelling their pets past level 31 in order to have the benefit of better skills and boosts.
Sometimes when you're in battle you may find monsters attacking your pet and making its 'vital power' level drop. Your pet's 'vital power' level (the red bar) is basically its HP. If you let this bar drops to zero then your pet will faint. You must then go to the Animal Trainer and select 'Unseal Pet' to revive it. Unsealing your pet will cost more per level so unsealing a really high level pet can get expensive.
Your pet's 'Degree of Starvation' is shown by the yellow bar, if this reaches zero your pet won't be able to move and will stop gaining experience from your battles.
There are two items you need to feed your pet in order to keep it healthy; green herbs and quality stones. Green herbs can be found outside any city in the herb patches. You must buy a knife from the general vendor and then select a herb to begin harvesting, this will take a long time unless you have some tool aides. You can apply a tool aid by clicking on it and dragging it over the tool you are using. Tool aides are found pretty commonly throughout monster drops in most temples.
You will also need quality stones to keep your pet healthy. First you need to by a mining tool, which varies depending on your class, from the general vendor, then go to a nearby mine and mine for quality stones to feed to your pet.
If you you find yourself without any tool aides and don't want to spend the time gathering the resources for you pet; there is a few players around the town centre that sell pet items but they are notoriously expensive. Try to but some tool aids if possible, otherwise try to put aside some time for resource gathering.