There are lots of trainers located all over the world that teach you many things. However, they can be a little hard to find at times and some of the trainers are actually hidden in obscure places. Here's a list of the various trainers ranging from Mounts to Weapons to Professions.
Weapon Trainers
Characters start out with various weapon skills already learnt at level 1, but that is hardly the full range of weapons your character is capable of wielding. To use weapons that you currently cannot use, you will have to visit a weapon trainer teaching the appropriate weapon skill and pay a small fee.
Weapon trainers can only be found in cities, once you have reached a city, finding the weapon trainer is just a matter of asking a city guard where he/she is located.
Once you found him he can either train you or advise you on where other trainers are. If you want him to train you he'll show you a list of which weapon types he can teach.
Mount Trainers and Vendors
When your character reaches level 40, he/she will be eligible to buy a mount. While mounted, you will travel 60% faster than normal. Each race has its own unique racial mount, for example, Trolls ride Raptors while the Tauren ride Kodo beasts. Before you can ride a mount though, you have to learn the riding skill specific to the mount first, such as Raptor Riding or Kodo Riding. The cost of learning how to ride is fixed at 20g, and the mount itself costs 80g.
Players can always learn how to ride mounts of their own race, to learn how to ride mounts of an allied race in your faction you have to reach the Exalted(?) reputation status with that race by doing quests and other things for that race, otherwise the mount riding trainer will refuse to teach you. Additionally, players cannot learn how to ride mounts of the opposite faction as the NPC trainers are hostile (and cannot be Mind Controlled). Due to the Tauren's huge size, they are restricted to only be able to ride Kodo Beasts and cannot ride other mounts. Any player can ride a Kodo though, not just a Tauren.
Mount vendors are located right next to the mount trainer itself. They sell mounts of various colours and also sells level 60 versions of mounts that travel 100% faster instead of 60%. However, these magnificent beasts cost 1000g to purchase. The Humans have 3 Horse Vendors while the others only have 1.