Anti bark collars are A Very Bad Idea. My DDog was handed over by his previous owners with a vibrating anti bark collar, and I suspect it was a significant cause of his anxiety problems, which will likely never be fully resolved. He effectively became too terrified to bark in the house, but the problems it caused have been much worse and harder to fix (eg he essentially becomes very loud and aggressive towards anything he's scared of. They're an aversive, punishment based tool and no reputable trainer would recommend the use of one.
What needs to be done is to identify the cause of the barking and remove it so that the dog no longer feels the need to bark. For instance, if the dog is barking when they're out, chances are the dog has separation anxiety and they need to see an APBC or CCAB accredited behaviourist so that they can work on the dog's fear of being alone. If the dog is barking at passers-by, then blocking his view (eg by closing the curtains) should improve matters.
Occasional barking at things like the doorbell and postman, or barking during a really exciting game out in the garden are normal dog behaviour and part of the fabric of life when you have neighbours (along with building noise, BBQs and the sound of children playing).
It is common for dogs to be unsettled by a house move. Mine became more territorial and started barking at anyone in the shared hallway and visitors. Thankfully I have nice neighbours, and I did mostly manage to resolve it but it took several months and professional help, despite not being an issue before we moved.
In the first instance, speak to your neighbours and ask if they are aware that their dog is barking so much, and take it from there. If it only happens when they're out, they may be unaware, or they may not realise the dog's barking is audible in your home. Assume they're reasonable people until proven otherwise 