Player training depends on two factors - your Training Centre and your coaching staff. For better training efficiency, you should have good coaches as well as a good training centre. You can access your players training status via Training page under the Squad menu.
Training sessions happen at the start of each round in the game. The player will train on the currently selected attribute during the training sessions, however some other attributes are also likely to improve but at a much lower rate than the selected attribute. Players usually train faster when they are younger, so it is wise to train their important skills early on. Also some players are faster learners than others and tend to develop their skills at a higher rate.
Player regain a certain amount of fitness every half an hour. But sometimes you need to rest a player during a training session to help him regain his fitness faster. Rested players spend that time along with your club's physios at the Medical Centre, and undertake relaxing treatment to refill their energy levels.
To instruct a player whether he should rest or do normal training during the training sessions, you can use the rest/train button (red/green circle). A green circle means he will train during the next session, and red means he will rest. There is also an option in the "Action" section at the top of the training page to assign multiple players at once to rest or train - all of them, none of them, or based on fitness.
Obviously, injured players do not train with your team. They resume training only when they are completely recovered from their injury.
Selecting Attribute To Train
From your training page, you can select which attribute you want a player to train on by clicking on the desired attribute. The currently selected attribute for training will be highlighted. You can also automatically assign an attribute for training to all players at once using the "Auto-set Training" button in the "Action" section at the top of the training page.
Players on loan at your club will still continue to train using the training facilities and the coaches available at their own club. What attributes they train will be controlled by their own club's manager, and only the owner club's manager can instruct them to rest during training sessions. Similarly, all players on loan will continue to gain experience based on matches played by their OWN team. This is to make sure that they do not lose out on experience gains in case they are not given enough match time by the club that gets them on loan.
There are 6 different training programs available to each of your players. The efficiency of these programs can be seen in your training page. The rate of improvement of your player in any attribute depends on the efficiency of the training program for that particular attribute.
Basic Skills : helps improve Passing and Heading
Goalkeeping : helps improve Handling and Reflexes
Defending : helps improve Tackling and Marking
Attacking : helps improve Shooting and Dribbling
Tactics : helps improve Positioning and Set Pieces
Endurance : helps improve Speed and Stamina
The best coach having expertise in each of the training programs automatically manages those, so make sure your coaches are good in each one of them to get good results. 50% of the efficiency also comes from your Training Centre level since you need the right equipments to train properly.