"I don't want training" sounds weird to me too, but I suppose if you think of training as something that a trainer does or gives to you, it makes sense. I can certainly imagine an overly confidant athlete saying "I don't need training". (Not just an athlete, now that I come to think of it - I've had to sit through numerous mandatory safety training sessions that I didn't think I needed!)
In English to train, to go training, and to be trained all have overlapping meanings. "I don't want to train" could be said by an athlete/trainee or by a trainer. I don't know whether that happens in Irish, though.
You're right, though, that in Irish, traenáil is both the verbal noun "training" and the transitive verb "to train". But I don't know if you can have a transitive verb in a tá ... ó sentence.
The audio in this course was recorded by a native speaker of the Connacht dialect.
More likely to be Mayo than Galway, to my ear.