Mr/Ms RIGHT

First thing about bonding is finding Mr or Ms Right. I mean the pet needs to choose YOU. Imagine bonding with the wrong significant other. Just doesn't work, does it?

HOW DO THEY FIND ME?

I guess you have to participate to some degree. Dogs and cats don't usually have the ability to get to you when you are ready to adopt a new family member. I have been hunted down by dogs dropped off by irresponsible owners (don't get me started on THAT behavior) and neighborhood dogs that just decided that their new address was mine. But usually it goes something like this: You go into the shelter and while looking into the cages, your eyes lock with some big, scruffy dog that WAS NOT what you had in mind when you came in, but you are now driving home with that lucky creature on your lap. Or you go to see a litter of puppies and one just won't leave you alone. They can't "talk" but they certainly can communicate.
I want to encourage people to examine their own lives before they decide on what animal generally fits them.
Breeds of dogs DO have certain characteristics that will surface eventually. Beagles do bark. A LOT. So, if your chosen one has any Beagle in them, expect to become used to the "sounding off" that Beagles do. Labs retrieve, so they often become fixated (and I mean that in the Obsessive-Compulsive kind of way) on something they can retrieve- such as the tennis ball, Frisbee, stick, etc. My suggestion here is to actively choose the thing they become OCD on. Something you will find manageable. Otherwise they will choose and it's so hard to get them to stop chewing those sticks (and getting stick stuck in their throats) or rocks (I knew a dog that broke MANY teeth chewing rocks) We like the tennis ball. Cheap, easy to throw, easily replaced.
So, get a hang of what breed characteristics fit with your lifestyle. Greyhounds DO NOT need a ton of exercise, are gentle and low-energy. Make great pets.
Some people like a little more "boldness". Dobermans are intelligent and loyal, and usually bold. Small dogs have more energy and it might be a better fit for a larger, less active dog in that small house or apartment.
Just things to think about when pursuing the new pet.

MY PITCH FOR CATS

If you work all day, come home late, go away on weekends, only have bits and pieces of time for a relationship (that's what we're talking about, after all) a cat, rather, 2 cats are for you. They are playful, self-reliant, love to cuddle and can be dedicated to their person, but don't need the daily walks, play, and general maintenance that a dog needs- must have in fact- or things can get ugly fast.
Why 2 cats? They will play together while you are away, snuggle and generally take care of each other. My daughter's cat and Beagle keep each other company, so that is fine, but our cats would be lost without one another.

So, do a little planning when choosing the general kind of pet you are looking for, and they will find you. THEN the bonding can begin.