It can indeed be difficult to find good dog care here.
(There are plenty of folks offering... let's just say, less-than-good dog care - unfortunately. You will have to do your due diligence on anyone you are considering hiring.)
The platform you have already found - Petsitting24 - is indeed the largest source of people offering dog care services. The trick is winnowing out the less-than-serious carers to find the responsible, trustworthy, capable folks who offer services that match your needs. Looking for care, interviewing different providers, should be a top priority as you plan your move.
The best way to use Petsitting24 is to write an ad detailing exactly what you need, what you can offer, and then start interviewing all who respond. To post an ad as an employer you have to pay for a membership, but a year's membership isn't expensive.
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Just to clarify - are you looking for a dog walker, someone who would stop by your home once a day to give your dog a 45 minute walk, or are you looking for something more, say a doggie day care set-up?
That your dog is small will give you more options, and if your dog is well socialized to other dogs, people, and children, well trained, well behaved, healthy, and largely quiet you should not have too much trouble finding someone. Or several someones, as you might need several sitters if you need daily care. (Rule of thumb among my dogowning crowd is that you should always have at least two 'back-up' carers in mind, as the turnover in the field is extremely high. Not to mention that chance of a private sitter bailing on you at the last minute is unfortunately also rather high.)
Many sitters restrict their clientele to 'easy dogs' - the real difficulty lies in finding sitters who are willing and able to care for dogs with special needs, either behavioral or medical. Hopefully you won't run into that hurdle.
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So that said...
If you would consider a doggie day care, one in the greater ZG area is VIPets in Morgarten (and Gossau):
https://www.vipets.ch/de/
They are set up so that the dogs live in a group, family style. Your dog would need to be good with other dogs as they have a fairly large number of client dogs at any given time. I have not used them myself, but I knew one of the original founders (she is no longer with them, though). A couple of EF dog owners have posted about their experiences with VI Pets (perhaps under the name VI Dogs, as it was previously called) so you should browse through the Pets Corner for comments.
Not so far from Thalwil, a friend-of-a-friend uses Jacky's Hundesitting:
https://jackys-hundesitting.ch/angebot/
I have no personal experience, so take this as info only.
The search terms you would want are Hort, Hundebetreuung, Tierbetreuung, and of course doggie day care. There are other businesses in the areas you mentioned that would offer a similar set-up.
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If you cannot find a business, the alternative is to hire someone as your employeee. This is how most of the sitters listed on Petsitting 24 will likely want to work. Even the private individual who takes dogs into his or her home, but is not set up as self employed or as a business, will likely need to be your employee, to ensure that social taxes are paid and insurance is in order.
BTW, if you hire someone privately, there are services such as Quitt.ch that will take care of the administrative end.
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Be aware that prices are all over the ballpark. If you need individual care, if you need specific skills, if your dog is other than 'easy', expect to pay up for that.
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The difficulty of finding good dog care ratchets up when it comes to holiday care, especially as everyone seems to want to travel at the same times. Many places ask you to book well in advance, some owners I know book a year ahead. So if you expect to need to travel without your dog, make finding holiday care a priority too.
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Don't be shy about asking every single dog owner you meet when out and about for recommendations - given how difficult it can be to find reliable dog care, no stone should be left unturned. Ask at your vet's office as well.
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Speaking of vets, off topic but nonetheless good info to know about the area:
We are spoiled for choice of good veterinary care in this part of Switzerland. If you end up in the Zug area, two practices I highly recommend:
MyVets in Cham:
https://www.myvets.ch
And the Ennetseeklinik:
https://www.ennetseeklinik.ch/site/index.cfm
Whoever you choose for your 'general priactice' vet you will want to get to know Ennetsee, as they serve as the area's 24/7 emergency clinic.
Another 24/7 emergency clinic if you end up on the 'bottom half' of the Zürichsee left coast is Marigin in Feusisberg SZ:
https://www.marigin.ch
You should also know about the Tierspital, the veteriary hospital of the University of Zürich. In addition to providing a general clinic, a wide variety of specialists, and 24/7 emergency care, this is one of the leading research and teaching centers in Europe.
https://www.tierspital.uzh.ch/de.html
One of the things I find so impressive about the quality of veterinary care here is that small general practices, the area clinics, and the Tierspital work together quite well, referring to specialist as needed. Medically, your pup is in very good hands in the greater ZG/ZH/SZ area.
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IIRC other dog care options for your general area have been mentioned through the years in the Pets Corner - a bit of browsing might suggest other options for you.
All the best to you and your four footed friend as you settle in to life in Switzerland!