Yes, your players can get injured. Normally they get injured during matches, especially if you play them when they are too low on Fitness. Rarely, players incur injuries during training too.
The chances of a player getting injured is determined by his Fitness, the ability of your team's Physio (explained in Staff section), and the current level of your Medical Centre. Always try to give a rest to your players when they go too low on Fitness.
You will get a notification when one of your team's players get injured. You can also see the details of the injury from the player details page.
Players can get suspended if they accumulate cards during matches. Details of suspensions are covered in the Competitions section under Game Basics of this tutorial.