This question comes up quite a lot, I would do a search of the forum threads for more opinions. My answer is: it depends on your situation, skills, and abilities.
On TCO: Spec7re is right that motorcycles (usually) have a shorter lifespan than cars, but this is not always the case. Many motorcycle engines are all clapped out after 30k miles, but others can last as long as a car engine, or longer. Look for bikes in the touring and sport touring categories, these will often go practically forever as long as maintenance is kept up. But these are going to be bigger, heavier bikes too, which may not make them suitable for beginner riders and urban traffic.
Since motorcycles are necessarily much simpler machines than cars, almost all maintenance and repairs are within the ability of an average mechanic. Being your own mechanic is a necessity if saving money is your goal because $500 for two tires and an oil change at a stealership is ludicrous. There is loads of free leaning and community support on the Internet. You need at least a couple hundred dollars in tools and a clean place to work. If you don't have a garage, make friends with someone that does. I see people rent storage units as makeshift motorcycle garages. This works in a pinch but is too spendy for the long term.
On safety: The statistics say that motorcycles are far less safe than cars. Can't argue with science. However, take into consideration the kinds of riders you see most often on the roads. Most are either pirates on cruisers or ninjas on sport bikes. They're into motorcycles either for the image or the thrill of speed. Or both. Safety is way down on the list of priorities. Maybe one out of every 10 other riders I see is rocking ATGATT and riding sensibly.
Maybe it's a false sense of security, but I feel very safe when I commute to work on my motorcycle. Compared to driving my car, I have fewer blind spots. I'm only blind to my immediate left and right and it's very hard for a car to get there without me noticing. I still check before changing lanes, though. I sit higher than most cars, and I can shift my point of view to either the extreme left or right of the lane very easily, so I can see further ahead. Because the bike is smaller, there is usually more space between me and other traffic, so there is more time to react to a malicious or inattentive driver. As well, the bike accelerates and decelerates quicker than a car and is much narrower, making for many more escape routes than a car has should things go sideways.
On practicality: If you're going to ditch your car completely, you need a bike that is practical. These are few and far between in the U.S., unfortunately. You'll probably end up with a standard bike and adding cargo capacity to it (in the form of panniers, top-case, tank bag, etc) or getting a touring bike with those things already installed. The NC-700X is a very popular all-around commuter and utility bike but it's fairly new so might be rather expensive, even if used. Don't discount scooters as well, the biggers ones have all of the power and fun of a motorcycle and also usually carry quite a bit of cargo.
I have a particular fondness for the vintage Suzuki GS line of motorcycles. They are plentiful on craigslist, cheap (under $1000 for a runner, especially this time of year), easy to work on, very reliable, and have a great on-line community. I own a 1980 GS850GL and a 1981 GS750E, but I'm selling the E next spring because I like the GL just fine and don't need two motorcycles.
If I were in your shoes, I would learn how to ride, get an entry-level bike, and ride it for a year before settling on getting rid of the car.
There is a lot more that I can ramble on about, but this post has already reached wall-of-text status a long time ago, so I'll just end it right about... here.