I don't even know where to start. Do you want large or small? Furry or not. Require grooming or not? A dog that can withstand temperature extremes or an inside dog? Do you want a dog that can be off leash at all?
All dogs can be trained but some are more difficult that others.
Two of "my" dogs are chow mixes. Both are loving and fuzzy, can stay outside, love cats, are good watch dogs and are generally great dogs. One is mixed with a Wheaton Terrier and his only problem is waunderlust. Terriers tend to like to escape and roam.
Lab mixes are great dogs all around and are very popular. The Lab is the #1 most popular dog in the US. They take some work and puppy stage can last a LONG time. But, they are generally great dogs.
Beagles are wonderful but stubborn and have an annoying bark. They are hearty little dogs.
Boxers are great but difficult. We've never had a Boxer that could handle doggie day camp.
I have a Pitt mix that is the most loving and sweet girl ever.
Goldens are always great dogs. I've never met a bad golden.
Another favorite dog of mine is a Husky. Sweet, loving, loves to exercise and play.
Any of the herding dogs are lots of fun, very smart and trainable and they LOVE to play. I love Corgies and Border Collies especially. They will herd your cats and children.
When you're looking for a dog that will be friendly with cats, look at what category their breed is in. Generally, (and this is NOT always true). If they are bread to catch things, they may not do well with cats. That can be the case with hounds and terriers especially - but, again, that isn't a rule.
Dog Groups:
Sporting
Hound
Working
Terrier
Toy
Non-Sporting
Herding
I could go on and on. I've taken care of so many dogs and different breeds over the past year, I feel I've gotten to know the general personalities of different breeds, but when you have a mix, you never know.
Start with
www.Petfinders.com or find a breed you like and go to a breed rescue.