Count cards ;-) Less metaphorically: while trying to get better, focus on what micro-optimizations you can do. While playing to win, have micro-optimizations in mind, but always keep your eyes on the goal, which is gathering the five cards and bringing them to a research center.
The strategic advice in other posts have looked good to me. I think I might be able to help with some general analysis. While it (mostly) doesn't suggest what plays to (not) make, it might help to have the structure and logic of the game spelled out.
Time and city cards:
The deck has 48 city cards, 5 event cards and (say) 7 epidemic cards (it makes the numbers rounder without distorting the big picture), for a total of 60. Of these, opening hands of 8 or 9 cards total are drawn, so on turn one the deck has 51 or 52 cards. With 2 cards drawn each turn, that's 26 or 27 turns (if you can start but not end your turn on an empty deck). With n players, the number of turns each player gets is 13.5 (n=2) or 8.66 (n=3) or 6.75 (n=4); for n > 2, one player gets one fewer turn than the rest.
Note that with four players and 6 epidemic cards, between the end of your turn and the beginning of your next turn you will very often have drawn an epidemic card, whereas you get two turns between epidemic with two players.
You draw two cards per turn. The time it takes to have a well-underway set (3 or 4 of the same color) is a function of how many cards you draw. Meaning it takes the same number of rounds, where a round is n turns, but more turns to get close to having a set.
In a 4-player game you draw 52/4 + 2 = 15 cards (13 at the end of your turns, plus your opening hand of 2), of which 12 (48/4) will be city cards. If the colors are evenly distributed it's 3 of each, so you'll have to rely on card swaps for the remaining two; more likely it'll be 2-3-3-4 or something close to even. Unless you're very fast, most likely each player will only be able to collect one color, and you'll have to choose who collects what around halfway through the game.
This also means that discovering a cure is likely to happen quite late in the game, and the special benefit of having a cure (better actions) is likely to be unavailable except for a few turns. Likewise, the benefits of eradication (which happen when you draw from the infect deck) is greater the earlier you eradicate, but it's likely to happen late or not at all. (It gives a greater per-turn benefit in later turns, but a greater overall the earlier it's done.)
On the other hand, with two players you have 52/2+4=30 cards, of which 48/2=24 will be city cards, or 6 of each color. The need for coordination is low, you will likely discover cures throughout the game and reap the benefits thereof, and eradication is often doable if you focus on it. In my experience you discard a lot in two-player games due to hand limit; after each cure is discovered, it's probably a good idea to count city cards in the discard pile and calculate how many are left of each color.
For three players, you get 51/3+3 = 20, among these 48/3 = 16 city cards, or four of each color. Most likely you can discover one cure without help, but a moderate amount of swapping will be useful. Also, who discovers the fourth cure? Having 8.66 turns, you probably can't afford to just wait for the cards to show up, but I haven't played 3-player all that much.
Roles versus the number of players:
- The Medic has powerful actions; the more actions he can take, the better. The more players, the fewer he can do that; his effectiveness is about 1/n in a game with n players—except for synergy with the dispatcher once a cure has been found.
- The Scientist gains more from drawing and receiving cards, and he draws card more often the fewer players there are. Receiving cards is made easier by the Researcher (less mobility is required) and Operations Export and Dispatcher (mobility is cheaper).
- The Contingency Planner can replay event cards, but only those that have been drawn. The probability that an event card has been drawn since the beginning of the CP's last turn is roughly linear in the number of players in the game, so he becomes more powerful the more players are in the game. He is enabled by other players.
- The Operations Expert has very flexible movement on his own turn and enables the fast movement of other players on their turns. His power on his own turns is independent of the number of players, but he adds power to each other player's turn, so he also gets better the larger the number of players. He enables other players.
- The Dispatcher is both enabled by having more players in the game (allowing more warps), and there is more need for coordination the larger the number of players. He enables other players, and is enabled to do so by the presence of more players.
- The Quarantine Specialist allows other players to not worry about a given board area. The number of "safe" turns bought by the QS is the the number of non-QS players. She enables other players not by adding to their power but by removing an expenditure of it.
- I'm not quite sure about the player-scaling characteristics of the Researcher: one the one hand, other players can take cards from you on their turns, which suggests it's more powerful the more players there are, but that's limited by two factors: 1. you only draw two cards each turn, so you can only give away two cards for every turn you have; and 2. you need to meet up with other players to exchange cards, which is hard unless the Dispatcher or Operations Expert can help. Also, a larger group of players have a greater need for the Researcher by the above section. On the other hand, with two players the coordination tends to be easy, so once the two of you have a set of five cards you can rapidly convert it into a cure, and you have longer time over which to reap the benefits of having a cure. Think of it this way: the Researcher has a joint up-to-14-card hand with each other player individually, but it takes longer to get to 14 the more players there are in the game. (~12 turns for n=4 vs. ~4 turns for n=2.)
More about the roles:
Of the four that I'm certain get better the more players there are in the game, the Operations Expert is great in a two-player game and the rest are weak. The OE is awesome in a large game, the Dispatcher is good and the QS and CP are sort of OK. (As others have said, movement is very expensive and the OE greatly cuts down on some of the costs.)
From the above you should not conclude that the best two-role team is Scientist and Medic, or that the best four-player team consists of OE, CP, QS and Dispatcher.
The Dispatcher has synergies with the Medic if a cure has been discovered; he also makes it easier for the Researcher to deliver cards.
The Operations Expert has a two-move combo: build a research center, then discard a city card to fly anywhere. This can often be followed up by one more research center and the removal of a disease cube; i.e. he can act as a one-man strike team and have the crew show up later. He can also facility card delivery, but this hasn't been a big part of my plays.
Together the Scientist and Researcher can hold 14 cards. If they have 13 city cards, there is one color they have at least four cards of, meaning the scientist can discover a cure if they swap stuff around. (It might involve discards, but then there's one less color to hold.)
Comparing Scientist and Researcher: the Scientist moves the 5-card finish line closer to the starting line. The Researcher lets you travel faster towards the finish line (2 players) and/or give everyone else a push forward (more players). Of the two, the Researcher is more effective, in my experience, which admittedly is mostly two-player games.
Comparing Dispatcher and Operations Expert: the Dispatcher makes everyone more mobile on the Dispatcher's turn, i.e. in one big burst every n turns. The Operations Expert makes everyone more mobile on their own turns.
Comparing Medic and Quarantine Specialist: they both fight disease. The Medic removes cubes from the board; the Quarantine Specialist prevents them from getting there. It's easier to plan how to use the Medic; it's more luck-dependent with the QS, but you can boost your luck if you keep track of what cards have been shuffled and put on top of the infection deck (and can come up in the future), and what has been subsequently been drawn (and can't before an epidemic shows up). That's incredibly demanding if you want to keep track of multiple shuffle-and-put-on-top steps; I don't know how big a difference it makes.
Unless you're at your hand limit, it almost always pays to hold the Forecast event card until you have drawn an epidemic. By then, you can plan how to work around the epidemic. Before then, your plans might get messed up by an Epidemic card. Also, it is more powerful early in the game, where the number of cards you can look at is three times the infection rate (late game it'll be 2 or 1.5)—i.e. where you can plan the largest number of turns ahead.
Similarly for Resilient Population: hold it until an Epidemic shows up and you have dumped three cubes on a city. At that point you have the most information, and know what you need to protect against.
One Quiet Night is more powerful the larger the infection rate it (effectively) reduces to 0, i.e. later in the game. But if you only get 3, or the 2 that saves you from an outbreak, it's probably still a good play.
I always play Government Grant straight away and near the disease or where mobility is needed (e.g. for a card swap), and have not regretted it. The future is always equally uncertain, so you might as well get benefit for the largest number of turns by playing it earlier. Likewise for Airlift.
When deciding whether to spend a card, remember that the Contingency Planner can pick it up later; plan up to and including his turn.
If someone needs to collect two cards and discover a cure on their turn, they have one action to spend on movement. This implies the have to start their turn in a city in which they can collect a card, and spend one move action to get to another city where they can also collect a card and which has a research center.
If they only need to collect one card and discover a cure on their turn, they can afford two move actions, e.g. one to the collection site and one to a research center, or two to the collection site if there's a research center in that city.
When drawing an epidemic card and drawing the bottom of the infection deck, the newly drawn card is shuffled and put on top. The Quarantine Specialist is the only role that can prevent an outbreak caused by drawing that card again. The outbreak can also be prevented with Resilient Population and plain ol' not-having-bad-luck.
The ordering of card draws in the infection phase matters: if you have two connected cities, one with three cubes and one with two, if you draw the two-cube city first you get 2 outbreaks, but you get 3 if you draw them the other way around.
Exercise (while playing): whenever you have two or more connected cities, if you were to draw a card for each city in the cluster, count (or estimate) the smallest and largest possible number of outbreaks caused by this chain of events. Experience report: a city with three cubes that's removed from the deck by Resilient Population can still cost you the game if it's connected to enough large cities. (We had 5 outbreaks that turn.)