I was quite unfit in high school and tried running in university. I've now been running a few times a week for the last 12 years, but only upped my runs to 10km+ the last 2 years. My fitness has improved by upping my distances, and it's had the added benefit of shrinking my perception of the size of the city for me.
To fit running into my busy schedule during the week I now run commute home from work 2-3 times (about 7km each time), and fit in another run on the weekend (13km). Running home is pretty liberating and only takes me 15 minutes longer than the subway (Toronto subways are very slow), so it's worth it rather than coming home and then running after.
I have a few tips for anyone new to running, that helped me but took me a bit of time to figure out (feel free to disagree with me):
(1) Try to force yourself into a regular breathing rhythm right at the start of your run, especially if there are a lot of hills on your route. If you're trying to improve your lung capacity, I've found upping your breathing rhythm early on, before you feel like you need to breathe faster, can help you from running out of breath later.
(2) Avoid heavy shoes with excessive amounts of heel padding (which would be most running shoes out there) and try either flat lightweight shoes designed for road races or more minimalist shoes (like Vibram FiveFingers or Vivobarefoot Evo/Ultra) so you can run more on your forefoot or midfoot, instead of landing on your heel. I'm partial to Vivobarefoot Ultras myself. The Nike Free 3.0 running shoes are ok, but still feel like running on heavy unstable platforms compared to the more minimal shoes. Nike Frees, and shoes like them, are better for working out than running I've found. If you try forefoot running in really low profile shoes, then ease yourself into it at the beginning. You may experience some nasty foot and ankle pain initially, but that should subside eventually as your foot muscles develop and you improve your stride. Alternating between minimal and traditional running shoes sometimes works for people too.
(3) If you're going for distances, try to keep your leg and body movements as compact as possible so you don't waste energy. Shorter, faster forefoot strides might help you run farther.
(4) If your feet get numb after running for a while, it sometimes helps to stop briefly and stretch your feet our a bit. Note this is a lot easier if you have flexible minimalist shoes on. Preemptively stretching your feet at the occasional red light can help too, especially when running long distances on snowy streets.
(5) Develop a stretching routine before your runs. When I have the space and privacy, I do some funny looking yoga/pilates-inspired stretches that I would never do in public but really help loosen up my legs. This helps a lot when running in sub zero temperatures in the winter.
(6) Incorporate some long, challenging hills into your route. Aside from the benefits of running up hills, it might help motivate you to improve so that you can make it up all of the hills without stopping.