Took my 2 and a half year old DD every week for a while. She tried toddler class but was too shy. Her childminder has also taken several of this age group.
The rink we went to the coaches do private lessons at around £6 half an hour, cheaper than Penguin hire, sounds like.
Sounds like your DS got on quite well if he managed holding onto your jeans, so you are off to a good start. For comparison, most under 3s can't make sense of the ice at all and spend their first session hanging by their hands, incapable of supporting their weight on their legs.
I mostly held DD's hands and skated backwards so I didn't fall on her. This is like being a human penguin! This is absolutely the best way to provide support for a beginner, so long as you are confident skating backwards.
The first thing I taught DD was how to get up again if you fall down. And to hunker down, knees bent, hands touching the ice, and move along like that.
We used thick gloves, and double layers of clothing. No helmet and I made damned sure that she didn't ever fall onto her head.
Take lots of breaks, just do 15 mins at a time, when you are tired, that's when you fall. And eat well - take plenty of high energy snacks.
HTH