What is the best way to introduce a new dog into a home that already has a dog?
If you are even 1% concerned that the two dogs could dislike each other or scuffle, then consult a professional dog trainer in your area to properly introduce the dogs. A first meeting between dogs is like a first date between strangers. If it goes wrong, it is hard to repair.
If you know the dogs will get along but just want to get them started out on the right path then take them on numerous walks together. Dogs who walk together get to know each other by moving forward and doing similar things (like sniffing a tree). These “relationship building walks” will help them bond.
Once they are comfortable on walks then allow them into the backyard but keep their leashes on. All toys/beds/food bowls/anything they could fight over should be removed. Calmly praise them anytime they interact. Once they are comfortable in the backyard or if you don’t have one, allow them into the house.
Keep interactions short and sweet. It is better that the dogs have a short and positive encounter then a long drawn out one where they scuffle at the very end. You can separate them with a gate, crate and rotate or even use leashes to tether them to opposite sides of the room.
Stay relaxed. If you are nervous when the dogs meet then they will be nervous. If you stay happy and upbeat then the dogs will be too!