Currently the only way to get injured in MMA Tycoon is during a fight. The are no injuries during training because it would cause a whole lot of problems for fight organizations if their fight cards kept getting disrupted by injuries caused during training.
If your fighter has been injured, you will see the number of days he is injured for on his profile page. You can send your fighter into a fight with an injury but it will seriously affect his mobility and cardio so it is not recommended.
The more punishment a fighter takes in a fight, the more likely they are to get an injury.
You may still train as normal with an injury.
Injured without getting hit
You don't have to get punched/kicked/whatever to get injured. You can get injured just as easily by punching someone as you do from getting punched. If you have been injured by brutalising your opponent, consider it an injured / broken hand.