First of all - congratulations on the new family member! Good on ya for adopting a dog in need.
I have a standard protocol when bringing a newly adopted dog into our family; some of what I do is because I have a multi-dog household, but most of it is applicable to any family situation.
There are three things one needs on hand during those first settling in days: A sense of humor, plenty of patience... and a well-stocked wine cellar.
The last is for those 'what have I done?' moments.
Because every single owner - first timer and old hand - has those moments. But then you look into those big brown eyes, and laugh.
In a nutshell: Take nothing for granted, make no assumptions, safety first - and start as you more or less mean to go on, setting your dog up for success.
Understand that everything is new to your dog. He doesn't know you yet, hasn't had time to suss out the lay of the land - so he might be a bit reticent, frightened or shut down at first, especially if he has been in kennels a long time. Or he may be over-exuberant, bouncy, excited at each new thing. It may take time - sometimes weeks, sometimes months - for the dog's real character to emerge.
Be prepared to roll with the punches, be prepared for a few surprises. (This is where the wine cellar comes in.
When I bring a new dog into the home I make no assumptions - ever. For the dog's sake, for the sake of all two and four-legged family members the newbie is always under my direct supervision during the initial settling in period - and for the few times when that isn't possible I set up a 'Newbie Den', a safe comfortable place separate from the rest of the household where he can relax or get away from it all, where there is no chance of conflict when I am not there to supervise.
I use a guest room off the main rooms - everything the newbie could want is in there - bedding, water, toys, etc. A baby gate separates this room from the rest of the house. The newbie can see what is going on, the other dogs can see him. Gradually I'll relax this, but in the initial phase it's either under my direct supervision or in the Newbie Den. Safety first.
Speaking of no assumptions: even if the newbie is thought to be house trained I start off as if he were a pup, not matter what age. Outside on lead with me, cue word, praise for performance, then release to play. I do this every few hours (depending on the age), just as I would with a puppy. Soon the newbie learns our routine.
I take NO chances when out and about at first. Walks are on lead only until I have proofed the newbie's recall. I use short lines in town, long lines in the fields - but never assume recall until you have trained it yourself. It only takes a split second to lose your new dog.
Prior to adoption I would have already gone through a lengthy assessment process to determine compatibility with my existing pack. But even then I don't take chances. The newbie has now come into the existing dog's territory so the potential for conflict must be considered. Therefore all toys or other resources that could spark conflict are put away. As above, the crew is always under my direct supervision. Unless I am right there, we separate until such time as the dogs have shown me that the newbie is fully integrated into the crew.
Obviously you will need to do a variation of this wrt your cats. I assume you assessed cat friendliness at the shelter? Even then, keep a watchful eye out with your cats, as this is their territory. If needed, keep the newbie on a house line until you have proofed cat-friendliness - even when you are right there. The house line means you can react quickly enough if there is undue interest.
Ditto children: Again, I assume your new dog was assessed as able to live with children - nonetheless, the children MUST have proper behavior around dogs drummed into their heads. No grabbing, no rough play, no running and shrieking - and no bothering the dog when he is in his sleeping place. Depending on the age of the children, you might want to instill a rule that they children must ask you first before instigating play. (Obviously you will be in a better position to judge if the dog is happy to play at the time.)
A dog must be allowed a 'time out' place - either a newbie den, or a bed in a corner, etc - somewhere where the children are not allowed to go, ever. Even the most child-friendly dog sometimes gets tired, and needs a break. As your dog is just learning the family routine, this is very important.
By the same token, even though I have assessed the newbie as compatible with my own crew I take no chances with dogs we meet out and about. If I don't know the owner we just politely walk by. Dogs and owners I know and trust are introduced to the newbie at our Hundeschule, a good neutral space which is fenced in, so that the dogs can bumble about off lead. (Remember we are still in the recall proofing phase, so no off-lead in unenclosed areas.) I watch body language carefully to get an idea of the newbie's reaction to other dogs - once I have a good idea of his character I might relax the protocol with strange dogs - or not, depending on what I learn in this phase.
Food is the ultimate resource that a dog might want to guard, so feeding is done separately during this phase. I usually feed as training rewards throughout the day rather than in a bowl, but this isn't a great idea with an established pack and newbie who has not yet shown that he understands the concepts of wait and share. So we go back to feeding in bowls - under my supervision, giving each dog space. If I have any thought that the newbie has food issues I feed him behind the baby gate in the separate room. Gradually as I learn more about the newbie's character I relax this - and try to switch over to reward feeding soon-ish.
In your case, be careful to feed the cats and new dog under supervision or separately to avoid food possessiveness issues. And the children must understand that they may not tease the dog with food - or even hold their own food in such a way that the dog might get the wrong idea and mug them for it.
You mentioned that this is your first dog - have you already done the SKN theory course? This is the course that is required for all owners who have not previously had a dog registered in their name in Switzerland; having grown up with dogs doesn't count, only having been the legal owner, with the dog registered to you does. The theory course must be done before you acquire a dog. If you haven't done it, do it ASAP. Your son isn't required to do the course, you are as the legal owner, the person in whose name the dog is registered in ANIS.
The newbie will obviously have to do the SKN practical course as well; again, this must be done with the adult in whose name the dog is registered. Some trainers allow another family member to accompany you, but check first. Understand that ultimately, the responsibility is yours.
Positive, reward based training is the way to go - reward your dog for the behavior you want, rather than punish that which you don't want. (All the SKN classes are based on positive training methods.) Dogs need their intelligence exercised as much as their bodies; a good mental workout is as much a daily requirement as a good physical workout. Dogs relish learning - we owners need to provide them the opportunity, every single day.
I start training the day the newbie comes home. It's preferable to do so in a class, but if that isn't possible I work with the newbie myself. Since my other dogs all go to Familienhund classes I would have already arranged with the trainer for the newbie to join a class well in advance of his arrival. It's preferable to start with the SKN first, but if there isn't a class for a few weeks or months I start with Familienhund on day 1 and do the SKN as soon as possible thereafter.
Some of my dogs have been in awful physical shape upon arrival; when that is the case I start doing training as appropriate for their condition. If a newbie is in such mental distress that Familienhund would be counterproductive at this time I ask the trainer to do 1-1 sessions with us at our home, or I work with the newbie by myself until he has recovered to the point where he can handle a more social setting.
But every dog should start training right away - because training is an excellent way to build the dog/owner bond - and it's an activity that most dogs absolutely love. This is the 'start as you mean to go on' point.
Because I have a multiple dog household training is two-fold. First I work with the newbie on his own, and then again with the group. Every skill needs to be proofed in a group setting as well as on the dog's own. In your household, every skill needs to be proofed not only with you, but also with the distractions of the cats and children around.
In addition to training classes, I work on social skills from day 1 - that means walks in town, taking the trains, stopping by a cafe for coffee, etc. I go at the newbie's pace, I never put a dog into a situation he might find difficult to handle - but this is on the top of the to-do list.
A couple of good all-purpose training books:
Dr Sophia Yin, 'How To Behave So Your Dog Behaves'
Jean Donaldson, 'Train Your Dog Like A Pro'
And a book I think every dog owed should read:
Brenda Aloff, 'Canine Body Language, a Photographic Guide'
Another life skill that is necessary is the vet visit. Your vet is the only person who can do the ANIS registration, so you need to have this done soon anyway. Let the practice know that this is a first visit, ask if a little extra time can be put aside so that the dog has a chance to look around, greet the staff, get a few treats - make sure it is a good experience. I prefer to do the introductory visit first, without treatment that might cause distress (like a jab). I want the newbie to have a good experience with the vet from the start.
I do not allow visitors in the initial settling in phase - largely because I can't control visitor's actions. Until I have thoroughly assess my newbie's training, social skills, anxieties, training needs and flash points I keep the house quiet, stick to a low-key routine. In your case, I would discourage you from having your children'f friends over as your dog is settling in - that's often too much activity so soon in the relationship. If you do have visitors, your dog should be under your direct supervision at all times - or in his safe separate place.
And lastly, practice good etiquette at all times. Even if your dog seems friendly and eager to greet others, always ask the other person or other dog owner if contact is OK before letting him approach. Not everyone is a dog lover, and not every dog wants contact with his fellow canines. Respect others at all times. Follow Gemeinde, canton and federal law, keep your dog leashed where so required, pick up after him.
Much of this is only common sense - as you get to know your new friend - and he you - make no assumptions, keep an eye on safety at all times. Managing the settling in phase simply sets your dog up for success.
Wishing you, your family and your new doglet all the very best.