"...Is buying a digital converter box a good investment, especially for my situation?..."
Maybe. It depends on your TV. If it's an old analog set, you'll need the converter. If it's a modern flatscreen (you bought it in the U.S. after March 2007), it has the necessary digital tuner built in. The converter does not apply.
"...Can you give me information about your experience with it?..."
I've set up and tested several different digital-to-analog converters, and they all do the job. But again, see the first answer above--you may not need one.
"...Is it a good deal? How much is one?..."
Digital-to-analog converters are generally available for prices starting around $30 and up.
"...Do I need an antenna with it? If so, how much is it? How many channels will I get?..."
Now you've hit on the heart of the matter. The antenna is absolutely necessary for over-the-air reception, with or without a converter. What antenna is right for you depends on your location. A full size model above the roof is always the best choice, but that's probably not possible for you in an apartment. You're probably going to be limited to an indoor antenna. That *may* get you a few channels in HD--again, your location is critical.
If your apartment is in or close to a major metro area with several over-the-air TV stations, an indoor antenna might do the job. If you want a recommendation on which one to buy, be advised that there is a hoard of near-worthless junk on the market sold as "HD" or "digital" indoor antennas. To make this story brief, there's only one I would buy if I had to get an indoor antenna (see the first link below).
If you want to dig into this subject further, put your apartment's address in either or both of the next two links below if you're in the U.S. Use the 3rd one if you're in Canada