Characters earn EXP from victorious battles which helps them level up and increase their stats. The game features a Training option that will allow you to select “Training Partners”. Training partners are basically “sacrificed” to the character you're training since they disappear after the session. Selecting training partners with the same color as the trainee gives bonus EXP. You can still use other-colored characters but you'll only get the default amount of EXP. The training partners' level also affects the amount of EXP (and cost) they can give to the trainee.
Selecting a Training Partner that's the same exact copy of the trainee (both in nickname and character name) will increase the trainee's Super Attack level. If the selected training partners are of the same type (color), the trainee will receive extra EXP as well. Training costs Zeni (game Currency) so expect to spend a lot of cash for training higher-level characters.
Training EXP can also be boosted with the use of Training Items. These training items can be acquired on story maps. You can also select the Training Location to further increase the EXP gain the trainee will receive. The default location is the Wasteland which provides a small amount of EXP bonus but doesn't cost anything and doesn't disappear after use.
To sum it up, training opponents, items, and locations (except the Wasteland) disappear after the training session.
TIP: To get the most out of your training partners, it's recommended to Z-Awaken a normal character at LV20 before "feeding" it to your trainee. You'll also get a whole lot more EXP if you're going to feed it to the trainee with the same color.