Greyhounds can be agressive, so just be aware of that. They are sight hounds, and those that have been raced (especially) have been trained to follow that hunting instinct.
Also, to those suggesting rescues, that does not preclude the need to research dog breeds. There are many purebred rescues available and mutts will still retain the physical and mental characteristics of their base breeds.
In order to find a dog breed that fits you there are a lot of questions you need to ask. First (and maybe foremost), what does your landlord allow? If there is a weight limit of 40 pounds that will nix a lot of suggestions right off the top. Also, if you are facing a weight restriction, make sure you keep that in mind if you look at a puppy from a rescue. Guessing a mixed breed puppy's origins can be a complete tossup, and there are many many cases of people adopting what are expected to be medium size dogs and end up with a 100+ pound adult dog. This isn't helped by the fact that, unfortunately, some rescue facilities will lie about a dogs age, breed or predicted weight just to get it out the door.
Next, be honest with yourself about how active you are. Dogs have some basic needs, and the top three are food/water, shelter, and exercise. Yes, they need your love and compassion, but if they aren't healthy and fit no amount of petting can fix it. If you love to run and want a companion for your workouts look at a medium to large breed that is known for its energy. If you're more of a couch potato who only enjoys a shorter daily walk then a small or less energetic dog is going to be right for you.
Consider your space. How big is your apartment? Is an energetic dog going to feel cooped up, regardless of you daily exercise? There are large breeds that can adapt well to small environments, and there are small breeds that need space. Consider the size of your space in addition to the size the dog.
Consider the environment you live in. Different dogs have different health requirements that change with the area they are in. For example, I have two westies, and westies are notorious for their allergy problems. That means I need access to premium foods that won't upset their stomachs. It also means that, living in southern california where every season is allergy season, it takes more effort (read: time and money) on my part to keep their skin in good condition. If you live in Phoenix, Arizona and like to be outside a lot (especially in the summer) a breed like a pug, or any other with a short snout, isn't going to be a good choice since it can't regulate it's body temperature as well as a dog with a long snout.
Look at things like known breed health problems: are you really prepared to handle that $1500 hip displaysia surgery if you get a large breed that is prone to it? Consider shedding. Is it a breed known for separation disorders (incase you need to leave it alone)? Is it a breed that is easily trainable or is it known for being headstrong?
To sum this up in a shorter manner, you first post isn't enough information to recommend a good breed or type of dog for you. There is a lot to consider and the first step to keeping you and the dog happy and healthy is being honest with yourself about your habits, expectations, and the amount of time and effort you are willing to put into it.
I'll spare you my advice about training since this post is getting really long. If you'd like my opinion, just say so.
Good luck!