The best place to train your Tems is the Melee Tem Shrine in Arbury. This is pretty late in the game and will require you to defeat the followers within the shrine. Once you do you will unlock the ability to fight level 100 Broccolems.
To level up as fast as possible you should choose one Tem that resists either Melee or Nature techniques. This Tem should also have a level close to or at 100, so you aren't knocked out as fast. My go-to Tem is Nidrasil as he resists their Nature attacks and has high enough Defense and HP to tank the Melee moves.
Equip your Nidrasil or other Tem of your choosing, with a Lure to ensure all attacks are aimed at them, and be sure to have your Tem in-training as its partner.
A Lure is a piece of Gear that re-directs all incoming single target attacks from wild Tems to the holder. This does not work for spread attacks, so your Tem may get knocked out a few times while it's at lower levels.
Learning aids are items that double the amount of experience your Tems earn for 30 minutes. This, along with your Tem gaining experience battles with two level 100 Tems will make leveling up super easy. Learning aids can be obtained via rewards from the FreeTem organization, the postal service aka Temtem Uber Eats, and from Lairs.