Hi Karl,
Welcome to the forum.
Your kids are very similar age to mine 11 & 16 so I have a little experience of what your going through.
Ok where to start
The very first time I took my eldest son shooting was about 4 years ago so he would of been about 12.
I had a pcp Theoben MFR at the time and this was the first rifle he ever shot, this was bench rested at my local indoor gun club range.
This was a bit hit and miss (no pun intended ) as I think he found it a strange experience and took quite a bit of time to get used to the scope settings and working out the ranges.
Once accustomed to the settings he found it fairly enjoyable but I had my reservations about starting him off on a pcp as he got used to the rifle pretty quick.
I got the impression he wasn't really that interested and left it as a simple taster session as his interests were more football and video games.
Fast forward a few years and about 18 months ago I got the Dad can I come shooting with you
So many trips to the club in late 2015 he was starting to show more interest.
I had a mix of rifles from springers to cheaper end pcp rifles which he was starting to understand the differences.
My plan was to get him his own rifle nothing expensive as he was mainly using my pcp which we were sharing at the time.
Eventually we decided to get him a Weihrauch HW30K (this will no doubt be recommended for you as an excellent lower power all rounder) as his own rifle.
My experience with that rifle wasn't very good and I won't dwell on those negatives I'll just say unfortunately that must of been a Friday Gun as usually the HW30 is held in high esteem on this forum by many shooters old and young.
My son enjoyed having his own rifle to use eventually it was reduced to being used on the shorter pistol range due to its issues but I felt a springer was perfect for him to learn better technique especially trigger control.
He also had chance to try many other rifles but really struggled with bigger adult sized rifles Air Arms Prosport was a nightmare for him way too heavy as the same with most other under lever type springers.
He was ok with a HW95 but would of been better with the smaller HW99
Over time he eventually got fed up with springers it got too boring for him manually single shot loading and he much preferred the multi shot pcp maybe this spoilt him but at least using the springers had taught him better control and trigger technique.
Co2 pistols and pcp rifles are his preferred fun choice although he's no longer a weekly regular the enjoyment is still there.
Sorry for the long story but sometimes it's better to have an idea of others experiences.
Personally I would start your family on a nice lighter type springer that you can all use and feel comfortable with.
Usually it's the pellet hitting the target that makes the most noise if back garden plinking so a quiet target setup would be a good start as silencers on springers aren't usually that effective.
There's nothing wrong with starting with a cheap used pcp setup there's plenty around if the kids don't want to go the springer route first but just be aware that going pcp before springer is usually harder work getting them to adjust to a boinger after shooting a dead action.
The biggest advice I can give you though is if you can use your club for advice usually folk are more than happy to advise and let you try out there setups etc