I have rescued two dogs and these are my tips:
Consider whether you really have the time, suitable home and patience to adopt a rescue. Do you have enough space, a place where the dog can be alone sometimes? A garden? Are you prepared for the potential medical and behavioural issues a rescue dog will most likely have?
Make sure you get as much background on your rescue dog as possible. Research the breed or mix of breeds if that is the case. For example, a terrier breed is very likely to view cats as prey. Not in every case, my ten year old adopted Jack Russell lived happily with my two cats but generally, not a good idea! Also look at breeds that will match your current lifestyle. So if you are not particularly active, dont get a high energy breed rescue.
Prepare children. Hopefully they already know how to approach dogs but if not, explain that it may take time for the rescue to warm up to them. To not lean over the rescue, to stroke under the chin rather than on the head and to respect the dogs space and things. If the children are very young, consider whether a rescue dog is really the right thing to do. Absolutely no dog is 100% safe around children so be sure you will always be around and have measures in place to monitor the dog and the children when together. Always.
Look at vets and insurance. As said, it is not uncommon for rescue dogs to have medical issues and certain breeds are more prone to medical issues. Also look into dog foods, what will suit your breed of dog the best.
When you first bring your rescue home, if possible try to have the first day and night with just you and the rescue. If the kids can have a sleep over with grand parents or if not, keep them away from your dog for the first 24 hours. If the dog has any comfort items from the rescue it came from, put them in the main room you expect your dog to spend their time in. Let the dog sniff and explore one room at a time, as much as they want or need to. Show them their food and water and where they will go to the toilet. Have items of clothing or blankets that belong to your kids around, so the dog can get used to their scents. Try to work out your dogs motivation style for behaviour, whether they are food orientated and so forth. Give lots of positive reinforcement with their style throughout the day.
When it is time to introduce the dog to the children, take the dog for a walk first. Let them burn off some energy and this will hopefully mean the dog is a little calmer and chilled out. With my children, I had them come into the room and sit on the floor, side on from the dog. Body language can be really important in how a dog responds to new people. They all had a little handful of my dogs favourite treat. It did not take long as all for my dog to be very interested in these tasty smelling children! Lots of treats, gentle under the chin strokes and he soon decided they were his new favourite people. I kept the initial meeting short, ten mins and then another ten minutes and so on until I was sure he and they were all happy to be in each others company. Knowing a dogs body language is also super helpful, to understand whether your dog is feeling overwhelmed or stressed.
My rescue dog was very reactive to other dogs when going out on walks. The rescue had warned me and I knew that his previous owner had dealt with it by just picking him up (small dog!). Knowing his breed and having experience helped me quickly get him able to walk without being picked up around other dogs, though he never did become fully comfortable with other dogs other than walking past them. I kept him on a short leash and was very aware of my own reactions when I saw another dog approaching. If i started acting anxious, then so would he. I had his favourite treats and kept feeding him them and walking firmly past with no real reaction to his barking. This worked well for him and walks quickly became much easier.
One thing I have learned from having rescue dogs, is that sometimes unwanted behaviours show up only once the dog feels settled in and secure. This can be a couple of weeks to a couple of months.
When it comes to leaving a rescue dog alone, try to avoid doing that for as long as you can manage to ensure the dog is settled, with a steady routine. Take the dog for a walk before hand and have plenty of enrichment toys to keep him entertained. A kong stuffed with treats for example. My house was always quite noisy with kids etc so I would leave the tv on for background noise. Make sure the dog has access to their safe space and water bowl.
My boy I adopted at ten years old was with us for 4 years before a health condition caught up with him. When I first got him, the rescue told me he wouldnt be able to live with cats or another dog and couldnt be walked or socialise with other dogs. By the time he passed on, he had lived with two cats, another dog he adored and only marginally wanted to murder other dogs when out on walks. Not bad for an old boy jack russell terrier. He was never a huge fan of my kids, that was my other rescue, but he accepted them and they respected his boundaries. He was my little shadow and I miss him very much!