Collins lists the first construction here as [verb + particle (on) + to-infinitive]:
5 If you go on to do something, you do it after you have done
Alliss retired from golf in 1969 and went on to become a successful broadcaster. [VERB PARTICLE to-infinitive]
She went on to say that she had discussed it with the Canadian foreign minister. [VERB PARTICLE to-infinitive]
It also lists a construction similar to the second, but as [verb + particle (on) + preposition (to) + noun (place)]:
If you go on to a place, you go to it from the place that you have
He goes on to Holland tomorrow. [VERB PARTICLE preposition/adverb]
A metaphorical broadening of this, and the use of an ing-form in place of the noun, are not unacceptable per se:
That's all for today. We go on to knitting cardigans next week.
But 'He goes on to saying that ...' is a poor, probably unacceptable, replacement for 'He goes on to say that ...'. This broadened sense (We go on to knitting cardigans next week.) can be paraphrased by, for example, We go on to look at / cover knitting cardigans next week. or We move on to knitting cardigans next week.