This is a bit longwinded, but should be helpful.
In general, the advantage of doing heat press is 1) you can do any number of colors (even photographic) and 2) you can print small runs and custom orders. Screen printing offers 1) Better quality (opinions vary here, but more say that screen printing is better - not by nearly as much as it used to be, though) and 2) the ability to print on to dark shirts. One final advantage to heat transfer is that it's MUCH cheaper to get started in if you want to print them yourself, especially if you can find a used press.
The quality of using heat transfer really depends on the quality of your supplies. Obviously a heat press is going to be a lot better than an iron. You'll want a good ink, either Epson's OEM Durabrite or Magic Mix (Epson printer highly recommended). You'll also need a good transfer paper like the Transjet II (a.k.a. Magic Jet). If you use quality materials, the shirt quality is nearly as good and long lasting as screen printing.
As for the amount of room a heat press takes up, it obviously varies on size. The smallest press you'd want would be a 12x10 (aprox) press than can only do 8.5x11 paper such as the Hix Hobby Lite. These have the advantgae of being relatively portable (35 pounds) so you can use them to do custom jobs at special events, for example. You can go up to the 15x15, which is an average size, probably the most desired, or even up higher to have 15x21 (heavy, 100+ pounds - this is what I've got, about).
What you'll want depends on how big of prints you want to do, as well as what you can find (I'd recommend a used one). Even if you're only doing 8.5x11, a larger press can be very handy for actually being able to position the shirt and center the transfer (this one I can say from experience). There are also two different types of heat presses: Clamshell and Swing-Away. Swing-Away presses are a little more expensive but are considered easier to work with and heat/press more evenly. Clamshells take up less space.
As for the acutal space used, the recommended 'working space' (including room to swing the press back) for the press I have is as follows:
Depth: 47" (119.4 cm)
Width: 37" (94 cm)
Height: 25" (63.5 cm)
Smaller presses will need less room, you can generally find the press size and the recommended working space on websites selling the presses. If you're really strapped for room you might want to go clamshell, otherwise I'd recommend a swing-away. As for brand, Hix is generally considered the best, Geo-Knight the worst because of it's electrical/burn-out problems (they'll send replacement parts and the presses themselves work fine, but it's an issue they've know about for years and haven't fixed).
As for where to buy a press, I'd suggest used. Call up local t-shirt shops (check the yellow pages under 't-shirt' and 'screen printing') and ask if they have a heat press they'd like to sell or if they know of anyone who does. That's how I found mine, and if you find it locally you can save the shipping cost and actually take a look at it (and test it out) before you buy it. You can get a used one much cheaper (aim for 3-4 times lower than the price of a new comprable one), especially if it's just been sitting around for a while unused (again, as was the case with mine; he decided to go full-time screen printing, we got the press for about 1/6 the price of a new one). You can also check thrift stores, pawn shops, flea markets, etc.
Let me know if you want any more info about any of this; I can measure the actual space we're using for our press if you want me to and give info on where to buy paper, etc.
Here's about what my press looks like (mine is an older model):
http://www.instagraph.com/221_large.jpg
