During the time you are making your way through Tyria, there will be many new and exotic creatures and animals that you will cross paths with. Some of these can be tamed! But you cannot tame the fully grown, adult creatures. The only way to tame an animal, is to get it while it's juvenile (note: only "Juvenile" titled animals and creatures are tamable/charmable".
This is because they need to bond with you as they grow up (essentially). You are going to be its best friend, so it can trust your every command.
In order to find a young version of a species, try to look in the more protected areas of where the older ones of that species surround or protect. To identify the young from the old, there are a few obvious visual aspects:
The young are generally smaller than the adults.
The younger ones are titled "Juvenile"
example: "Juvenile Snow Lynx", "Juvenile White Raven", "Juvenile Alpine Wolf".
The young look less fearsome (their cuter).
When you have located the Juvenile version of your species of choice, it's time for the easy part!
Locate it, hunt it down, corner it, trap it and then finally tame it! (cast your "Charm" ability onto it by pressing "F", this ability will only be usable on tamable creatures)
*You begin to tame the juvenile animal -> you successfully tame it."
You have now tamed and unlocked your first non starter pet, one of usually many to come.
Now that it is your friend - seal the bond by naming it! Lets name it Mittens, Mittens the canyon spider.
Now that Mittens is tamed, you should begin leveling with it!
...oh and do not worry, pets in Guild Wars 2 do not need to be fed. Yay! I should also mention here that ranger pets are unlocked. This means, once you tame a animal, it is permenantly unlocked and yours to use when ever required. You can switch among inactive pets while out of combat. All your unlocked pets level up at the same time you level up.