I am and my wife are both instructors. We started bringing (or, better, PUSHING) our sons underwater (in the pool) around 6 months, leaving them to swim up. They never drink water, as small babies still have the mammalian reflex closing their epiglottis. Around 10 months both our sons were "swimming" using small fins and inflatables on their arms, and at one year, when they were not walking yet, they were jumping in the pool head-down and with open eyes after dog-walking towards the water.
All this activity was done in a special pool for babies, with very warm water, and with one instructor (either me or my wife) taking care of the baby, while an assistant was surveying, ready to intervene. This activity was organized by the local diving club, with proper insurance, medical clearance, etc..
Around 18 months we gave them mask and snorkels, and they started controlling their mouth breathing, diving underwater for catching plastic animals and other toys from the bottom of the pool, and clearing the snorkel when emerging. They did also start equalizing, albeit at that age it was not easy to explain.
At 2 years they had their first small air tank. The kicking technique was already quite good, and they had enough thrust for reaching back the surface despite the weight of the tank. They also had experience snorkeling in the sea, but not yet free diving (for safety reason we did give them a buoyant shorty suit and no weights, impeding them, at the sea, to dive more than 1m, although of course they were trying to go deeper).
Around 3 years they were "ready" and they used for the first time their small tank in the sea, at a maximum depth of 3 meters. They did already proof to always exhale when emerging without reg, and to equalize properly. They did learn to clear water from the mask while doing these short dives at the sea, in Sardinia.
Everything very very slow, one progress after the other, one week after the other. At 5 years old they both had something as 12-15 dives in the sea, and possibly 100 or even 200 hours breathing underwater in the pool and making any kind of exercise and playing games. For example, the instructor of the club did mount for them a plastic underwater castle, so they started getting used to overhead environments, both freediving and with the small tank.
So, at 6 years, again in Sardinia, at Capo Caccia, they had their first experience with small caverns and passageways. At the beginning vary shallow, and no real risk, but in a couple of years they could go in real small caverns, and dive down to 7-8 meters with confidence.
Here you see one of them inside a small cavern, at 7 years:
We continued teaching them very slowly. We did allow them to get certified only at 12 years, and after following a real OW course at the diving center of the Hotel Capo caccia, in Sardinia.
Then we went for one month in Australia, doing a 3-days LOB trip in Cairns, and after 10 dives, they were certified AOW.
However, I, and particularly my wife, were absolutely NOT happy of the PADI training they received in these two diving centers. So the following year we inscribed them at a real CMAS course, here in Parma, lasting three months and with real exams at the end, so they are now also CMAS-certified.
It is very important that children are trained properly. Slowly. Carefully. Leaving nothing to the fate. By instructors with proper experience and certification for small babies (my wife is enabled to teach since three years, but the instructor at the local club which did follow us during earlier years was certified for teaching to children since 6 months up).
Can all this be cheap? No, if you rely on a commercial organization such as PADI. In our case, it was incredibly cheap, as most of this activity was organized by our diving club. Once parents pay the annual association fee (which includes insurance), you have unlimited access to the pool, and these courses for babies have just a nominal fee of 50 euros every 3 months (there are two of them per year).
The activity at the sea was also very cheap, using our own small Zodiac MK-2 inflatable boat and our own equipment. Only cost was filling the tanks, ridicolous...
Even certification was quite cheap. The Italian diving center did make a good discount, as we were filling our tanks at them since more than 10 years, and they did see my sons progressing from small babies up to youngsters. And Pro Dive Cairns did in practice give us the AOW certification for free, being done within the 10-dives LOB package.
Conclusion of this long post: yes, it is far better if your children are properly certified, but first of all they must be properly trained. As you are NOT an instructor, you cannot do it yourself entirely (albeit you can provide some good basic stuff). And albeit I and my wife are instructors, in the end we inscribed our sons to a final real course (CMAS, not PADI), and with a well renowned instructor, who did not know them in advance, so they had to follow his authority. Not always parents can really push their sons to do something...
Last point: the time spent in water with my sons was by far the best time of my life, and possibly also of their ones. We continuously recall funny or dangerous moments happened to us during the last 30 years. Intense moments. So all the effort and the money and the time dedicated to this was fully repaid by these emotions.
We also did teach our son other "dangerous" sports, such as downhill, alpine ski, motocross, trial, free climbing, etc. But diving had definitely been the most intense experience.
So, please, do not try to save some money with a super-fast certification, lacking some proper good training. Allocate the time and the money required, it is all fully worth the result!