Original Game Edit
When your begining the game, the fastest way of training is Mousefur or Dustpaw, depending on who is located in the Training Hollow. Dustpaw drops Juniper Berries (usually 2) when defeated, so it's a good strategy to get lots of SP back.
When you're further in the game, train in ShadowClan territory. They give 1418 EXP and 812 acorns, as well as dropping antivenoms.
After Bluestar dies, train in the place where she died. You will not only find RiverClan cats, but ShadowClan as well. If you're lucky to get large groups of cats, they can give you a ton of EXP and acorns.
The New Prophecy Game Edit
When you're beginning the game, the best method to gain EXP fast is to head north from Sunningrocks and train with the RiverClan cats there. The good part about it is that Sunningrocks (just a map south from the training location) has a spot where you can fully heal your entire party.
When you're at the Tribe Cave, train with the Tribe Cats. An effective way of doing it is to have -claw have Claws of Destruction, so you can defeat them all at once. It is also recommended to have Claws of Destruction before setting off towards the Sun Drown Place, so that you'll have an easier time during the journey.
The Power of Three Game Edit
Once you've defeated the fox cubs for the first time, you can keep coming back to essentially re-fight them. They may be hard at first, but they give a lot of EXP, as well as having the convenience of being close to the camp.