Sore Losers
This game is awesome. Seriously. It's frustrating, but once you know what to do, it's rewarding. Just keep on trying. But if you really get stuck:
1) Get your infectivity and resistance up to 3 each.
2) This next part depends on whether you're in an isolated region to start with (Greenland, Australia), or a land-locked one (Europe).
a) If isolated, infect the whole population, then infect the airports until you get another country infected.
b) If land-locked, infect half the population, then infect the water supply. Move to an adjacent country and infect.
3) Repeat step 2 as fast as possible, infecting half-infected countries fully every other turn. Once you have all the continents fully infected, feel free to crush their hopes by impeding their cures and increasing your lethality.
This walkthrough is not foolproof, mainly due to the randomization factor. Don't get discouraged if you don't get every country infected on your first try. At least you have a general sense of what to do now.