Specifics:
If going tradition, there's a policy that gives culture spot and another that gives scientist spot, what will you choose? In every social tree there is a policy that grants more culture, so pick it ASAP. You'll get the scientific policy eventually, probably faster than if you just pushed for science, and your culture will stay strong.
When founding a pantheon, you can try to get the best from the resources you can work, but if in doubt, it's usually better to pick the pantheon that will give more culture.
To befriend CS early you need: to find them, to have a little army ready with horses if possible, then wait for the barbs attacking your CS neighbours and defend them.
Some luxuries grant culture, so go and reach and connect them. The same applies to Natural Wonders.
Build wonders. Most of them gives extra culture. The exception is if you're going full domination.
Some uniques give culture too, see the moais for polynesians, that you can build really early.
Even before caravansaries, you can improve your trade power just by building roads. They may seem expensive, but keep in mind that some routes are not possible if you don't set the roads. If your cities have roads and the other civs cities haven't, chances are that most of the land trade goes to your cities, bringing gold and knowledge, and spreading your religion if you have any. Not that this helps to be the tech leader, but it isn't a mid-game strategy. EDIT: You don't need to finish most roads, only the ones that serve to connect your cities to your capital, but this kind of roads I'm talking about are meant only to reach farther, so they are good if they just help to travel faster over rough terrain.
For expansion schedules, I'd say that it depends strongly on your neighbours, and your chosen starter policy tree. With Tradition you can keep just 5 cities and call it a day (you'll need plenty of wonders in this case), Progress allows more cities, and Authority is perfect if you are always pushing borders. Fighting isn't usually good for your tech progress, unless you're an authority follower, and expanding provokes wars. If you manage to develop your latter cities, they will be worth the extra science cost, but if you have to fight to settle there, it may not be worthy. If early game you get too behind in science, probably you expanded too fast. I try to be near 10 happiness before attempting a new city. To work with this, I may refrain my cities from growing too much if I have space to settle, so my happiness doesn't fall. EDIT: Again, try to be always at +10 happiness, for it gives more bonuses to production (check the happiness guide). Try to be always at positive GPT, for it stops your science production when negative.
When you are overrun by another civ, you need to see what their strengths are. Maybe they have uniques that helped. Some civs perform very well early game, only to fall in the hands of a warmonger. Others expand too strongly and fall behind in science the next age. Others seems unstoppable until you realize their strength is in trade and you can pass a trade embargo on them. Playing scientifically is, as for now, playing a time victory, so your main strategy should be to watch others, and put stones under their wheels.
