During the *Batman Beyond* series, there's the episode *[Terry's Friend Dates a Robot][1]* during which you can see Bruce Wayne actually training him to fight against robots, in a simulated fight against some of Bruce's old rogue gallery (The Riddler, Two Face, Killer Croc):

https://www.youtube.com/watch?v=ymkAIVIortk

Also, in episode *[Curse of the Kobra][2]*, Bruce also sends him into meeting with [Kairi Tanaga][3], whom Bruce knew from when he trained with [the same master][4] before becoming Batman, so that Terry could learn to fight more efficiently:

https://youtu.be/0NQGfr4xRcQ?t=163

It's more than likely that Bruce would continuously train him into becoming a better Batman, in order to increase his likelihood of staying alive.

It's questionable whether or not he became a master in martial arts like Bruce was, but we get a glimpse of how easily he took care of [Shriek][5] in the _[Justice League Unlimited][6]_ episode titled _[Epilogue][7]_, when he became a grown up man, long after the timeline of *Batman Beyond*:

https://youtu.be/Vd2qNR4phOU?t=56

On a more trivial note: there used to be a video on Youtube showcasing how Bruce (as Batman), always took out enemies coming from behind him by hitting them with his fist (which the video called his signature move). In a *Batman Beyond* episode, as Bruce was tending Terry's wound, the latter complained that he doesn't have eyes in his back.

Later on, in an unidentified sequence, Terry (as Batman) knocked out a bad guy who tried to jump him from behind, showing he actually learned from Bruce.


  [1]: https://dcau.fandom.com/wiki/Terry%27s_Friend_Dates_a_Robot
  [2]: https://dcau.fandom.com/wiki/Curse_of_the_Kobra
  [3]: https://dcau.fandom.com/wiki/Kairi_Tanaga
  [4]: https://dcau.fandom.com/wiki/Yoru
  [5]: https://dcau.fandom.com/wiki/Shriek
  [6]: https://dcau.fandom.com/wiki/Justice_League_Unlimited_(animated_series)
  [7]: https://dcau.fandom.com/wiki/Epilogue