Surprisingly, training for contingencies did not increase after Apollo 13. It was more important for astronauts and support crew to follow the same general approach of recognizing problems, determining their cause, and then applying a solution. There was little chance that the specific problems of Apollo 13 would manifest themselves in exactly the same way, and that reflexively following a checklist might not be appropriate in most future situations.
In fact, the mission was cited many times as justifying the importance of having consistent, standardized astronaut training. For example:
The increasing effectiveness of standardized crew training for operational mission aspects, the continuous addition of crew experience, and the greater spacing between launches permitted the crews of the later science-oriented missions to devote 30 to 40 percent of their time to the development of, and training for, lunar orbital and lunar surface science procedures. The effectiveness of the standardized training program was dramatically demonstrated during the aborted Apollo 13 flight. Furthermore, mission results showed that substituting the backup Apollo 13 Command Module Pilot for the prime crew Pilot 2 days before flight was practical and effective, even under conditions of stress.
Apollo Program Summary Report, p. 6-1
There was barely enough time to train for normal activities. As evidenced in the first sentence of the above quote, scientific content increased as the program proceeded. Given the choice between preparing for normal scientific activities versus myriad, specific, unlikely contingencies, precious training time was devoted to the former.
Having said that, I found three changes to training after Apollo 13.
Two more reference points were added to the telescope/sextant alignment procedure:
Crew training for sun/earth and sun/moon alignments in the simulators should be emphasized to handle contingencies such as occurred during Apollo 13.
Apollo 13 Mission Report
Prior to the accident, they tried to hold television broadcasts during other critical events. This did not go well, and was changed for future flights. (The situation was unrelated to the accident.)
The television presentation during the midcourse correction maneuver, as well as during transposition and docking, interfered with normal operational functions to a degree not seen in training. The lunar module pilot was forced to spend full time adjusting, pointing, and narrating the television broadcast. A suggested alternative for telecasting during dynamic events is to have the ground do all commentary. Crew-designated television can be conveniently performed during a lull period when full attention can be given to presentation requirements.
Apollo 13 Mission Report
According to the Apollo 14 Mission Report, photographing the impact site of Apollo 13's S-IVB third stage was the highest priority for lunar orbit photography during Apollo 14, and the CMP was given relevant training. However, this would have happened anyway, regardless of the Apollo 13 accident.
It is also worth mentioning that the Apollo Program Summary Report mentions several other missions impacting training, but says nothing about training changes due to Apollo 13.
The Apollo 12 crew devoted more training time for extravehicular activity preparation than did the Apollo 11 crew; and, because of a very detailed high-fidelity cabin-stowage configuration, both crewmen prepared for egress in a rather routine fashion.
Except for a temporary loss of inertial reference on the Apollo 12 mission because of the lightning strike, the primary navigation system enabled the crew to monitor booster-steering performance throughout the launch phase and to confirm satisfactory orbit insertion conditions. [...] The flight-crew backup steering mode was included in the training program because a less precise orbit was preferable to a launch phase abort in case a launch vehicle platform failed.
Also, the preparation checklist was changed to eliminate communications checkout problems encountered during the Apollo 12 mission.
Even later missions, such as Apollo 15:
On the Apollo 15 mission, the first of the 3-day lunar stay missions, the crew found that, with donning practice in the 1/6-earth-gravity environment and the confidence developed in extravehicular mobility unit performance, egress preparation times were consistently shorter than planned. Later crews confirmed that preparation times were considerably shortened after the initial extravehicular preparation.
and Apollo 16:
Because of an open circuit in the secondary gimbal rate-feedback loop during the Apollo 16 mission, the lunar-orbit-circularization maneuver was delayed, causing a major change in the crew procedures and mission time line. As a result, onboard techniques for troubleshooting this kind of malfunction were incorporated in the Apollo 17 training.
Yet no mention of 13's problems impacting training.