There's a couple rules to this process and I'll list the first DO NOTS
1. DO NOT introduce your new dog to existing dogs in the home or neighborhood dogs for the first 2 weeks your new dog is home.
In the first few weeks your new dog needs to learn house rules, get training and bond with you. DO NOT under any circumstances allow the new dog to meet other dogs on leash or off leash.
2. DO NOT take your new dog on outings to public places until AFTER you've established solid loose leash walking skills, being neutral in public and a solid recall and place command amid distractions.
3. DO NOT let your dogs eat together
4. DO NOT let your dog "meet" other dogs on leash ever.
Your new dog is NEW to your home and family. It doesnt matter what the dog's history was because the most important history starts on day 1 of bringing them home. So you have to establish a routine in your home first and start training on day 1.
5. DO NOT allow friends and family to visit your new dog until you've established a home routine and have begun training. Keep in mind it takes a dog 2 weeks to fully relax into a new environment. The first week or so a dog will suppress its behavior because its in a new place. Week 2 is where you may start to see problem behaviors that previous owners didnt know how to work on. This is why establishing a routine from the beginning will be paramount to success with your new dog.
Now that you've read what you're NOT supposed to do, here is a best practices for bringing a new dog home.
Weeks 1 and 2 | Crate and Rotate | This means your new dog should be sleeping in the crate at night, eating his meals in the crate, he or she stays in the crate while you're gone. You develop a routine where your new dog has structured time out of the crate with you only. If you have existing dogs in your home, they are never to be together in the first two weeks.
When one dog is out, the other is in the crate. No exceptions. You should also have a few lessons with a trainer scheduled so that you can learn how to properly train your dog.
Week 3 | By this time your new dog should be well settled in and should know house rules, basic manners and is comfortable walking calmly on leash. Now its time to start the INTRODUCTION PROCESS.
You will need a second person to help, this is where a dog trainer will be immensely helpful. Depending on both dogs' behavior and temperament, a walk together through your neighborhood with each dog with one person would be ideal. Taking both dogs to a park to work on ON LEASH obedience is ideal. You should plan to do these sessions at least 3 - 4 times before trying any off leash obedience. Again, this should all be done with two people (one person for each dog) and the dogs DO NOT interact. They merely rehearse known Obedience behaviors in the presence of the other dog.
Week 4 | Off Leash Obedience in a fenced area practice. Now that both dogs understand how to pay attention to a person while the other one is out, now its time to take them to a fenced area, this can be your yard or a dog trainer's facility. Place bed training will be key as this is where you would put each dog on a place bed and work with one dog's obedience while the other one waits its turn on the place bed. If both dogs are being neutral and in a relaxed state of mind, you can finish the session by allowing the dogs to sniff each other OFF LEASH. Be ready to end the session when both dogs ignore one another. This may only take 5 minutes.
If dogs start playing, be sure to not let them get hyper aroused! Practice recalls away from play, practice going back to place bed etc.
You will still be doing crate and rotate in the home
Week 5 | If all is going well in week 4, now you can start to practice off leash obedience in the home in short sessions maybe 2-3 times a day.
The dogs will be loose in the home only when you are focused on them and they are under supervision. You want to discourage playtime inside the home. The home is place to relax and calm down, not play. If they start to play, they go in a crate.
Week 6 | Now its time to allow the dogs more freedom together in the home. Also walking them together without the assistance of another person if possible. You will still want to make sure the dogs are crated separately when you are gone and when they eat. For more advanced training and obedience, you will need to work with a trainer to establish these advanced behaviors correctly. Everything depends on the dog and you will need a trainer to help you.