You have a dog but you'd like to have another dog for both your family and as a companion for your #1 dog. Great idea! I am a big fan of having two or more dogs that is if that is what you and/or your family want and not just to keep #1 out of your hair. I believe dogs deserve a great human family and a great canine family.
One of the most common reasons I deal with when people want a second dog is that their first dog has some form of separation anxiety. Very few anxious dogs are "fixed" by adding another dog to the family. Typically a dog suffering from some form of separation anxiety is suffering because of your absence so adding another dog to the scenario isn't going to fix anything. Seek professional help in this case before you contemplate adding #2.
So let's say your #1 dog is well adjusted and you just want another dog to come into the family, you have a big heart and more than enough love to go around. The first thing you need to consider when thinking about adding a second dog is who exactly is your #1 dog? What sort of temperament do they have? Where do they fit in amongst your family? It is imperative that you choose the second dog to mesh with the first dogs personality.
Generally I advise going with the opposite sex, it is usually a safe bet and even if they do have their differences things run smoother with a male/female pair. Some breeds are very dominant by nature and adding another of the same breed, even the opposite sex might be more than you want to deal with.
Matching physical size is something to think about as well. If you have a huge beast of a dog who is high energy and loves to play rough you don't want to bring home a fragile type who breaks easily. But you also may not want to go with another beast sized dog when picturing two beasts running around your home.
And hair, are you happy with the amount of hair you deal with on a daily basis? Could you handle double the hair or would you prefer to not add anymore additional hair? These are all things that you should think about.
Age of #2? Do you think your dog gets along well enough with other dogs that bringing home an adult dog would work? Or should you consider a puppy who is less threatening than and adult? Many dogs get along great with other dogs but have a difficult time with puppies, especially if they are up in their senior years and cannot enforce rules with the new little rascal.
No matter who you bring home as an addition to your family think first about your #1 dog or existing pack. You do not need to go to extreme opposites when choosing an additional dog as that can backfire as well. Adding a very dominant dog to a home with a very submissive dog can ruin the first dogs life as they know it. Just like bringing home a calm sedate type dog into the life of a high energy, super powered dog that can't stop. That can cause too much stress for the second quite dogs life.
But when you do bring home another dog, don't expect it to be love at first sight. If it is then you are truly lucky. Often it can take up to several months for dogs to mesh. When I brought home Luke almost 7 years ago Tilley did not approve and wanted nothing to do with him. She set things straight right off and let him know she hated him and didn't want him around. But with enough supervision and feedback from us she slowly grew to tolerate him, accept him to now although she'd never admit it, love him. This all took over two months, two very rough months.