To get tooth decay you need susceptible teeth, bacteria (plaque) and sugar. use fluoride to make teeth less susceptible, clean teeth to reduce plaque, watch your diet to reduce frequency of sugar intake.
You need to read labels on all food packaging. Anything with any added sugar (look out for glucose, fructose, sucrose, anything - ose, any kind of syrup eg corn syrup) will rot teeth. Beware of some things that look savoury but are full of sugar (tinned soups, baked beans). Dried fruit is a major culprit as it is not only sugary but very sticky too. Remember those children's cereal bars are sweetened with fruit juices. Sugar basically.
Drinks are also a mine field. I would restrict juice and fizz to meal times only, including squash, diet / sugar free. Diet / sugar free are still acidic and will promote if not actually cause decay, and will cause erosion (often harder to repair than decay). Drinks between meals should be water (not fizzy), milk or unsweetened tea / coffee (maybe not for a 6 year old though!).
Unfortunately you do need to be strict with the no sugar between meals rule. I would suggest that treats are saved to be had as "pudding" and not eaten between meals. If snacks are required offer something savoury: toast with marmite, a piece of cheese. I always worry that some of those fruit - flavoured rice cakes are too sticky and sweet: DS has the marmite ones instead.
I would suggest a check up after 3 months this time, rather than 6 just to reassure and double check. 6 monthly fluoride varnish application is ideal. Disclosing tablets used at bed time to target brushing properly. My rule of thumb is that a child is able to clean their own teeth unsupervised if they can bathe and wash their own hair alone. This is usually at least aged 8, often much later.
Make sure where possible any regular medications are supplied in sugar free formulations. Communicate with anyone who looks after your child regularly to ensure they understand the rules....how much of a treat is a sweet when it results in dental treatment?
Don't feel ashamed or guilty. These things happen. I'm sure that you will take the appropriate steps to change the habits to stop new cavities forming. I have seen children with very decayed baby teeth go on to have completely decay - free adult ones. Also, please don't be disappointed if you find that the fillings done today need replacing in due course, once filled teeth can be harder to clean and more vulnerable.