If you feel comfortable with Prince then you definitely have some idea what you're doing, but here is some early game advice to help you move on to Emperor (king is barely different from prince). This advice is quite unorthodox and unlikely to work in Immortal+, but I find it to be very effective.
1. I prefer taking liberty and strongly recommend you choose it over tradition. It provides a free settler, free worker, free golden age, and free academy. Besides the bonuses of this policy, the free settler allows you to skip ~12 turns of building a settler in your capital, the free worker saves $300 or 24 turns in your second city, the free golden age gives you enough gold for crucial early defensive unit upgrades, and the academy (from free great person) gives a science boost amounting to 1000+ beakers through the game.
2. To get ahead in tech early on, I prefer to do what I call the 'GL slingshot'. Basically, if it takes less or equal to 18 turns, build the Great Library in your capital. Take philosophy as your free tech, giving you a massive science boost (free library and +3 science) as well as opening up the national college.
3. Build order; scout, worker, monument, granary if there's wheat, GL, archer x2 or 3, depending on your neighbors.
4. Build your 2nd/later settlers; don't buy. Don't be afraid to settle more than 3 cities if happiness/time allows. Settle cities in either grassland/hills or flood plains/hills; plains are suboptimal, marsh is bad (do it if you want to grab resources, though), and tundra/ice is a big no-no. Try to make sure a city can reach at least 1 lux and at least 5 total resources by the 3rd ring.
5. Research order: pottery, mining, writing, archery, calendar or animal husbandry (depends on terrain), philosophy free, general beeline to construction (c-bows), general beeline to currency/civil service for gold and food. Whether or not you go education or machinery next, again, depends on your neighbors.
6. By the medieval era, keep a standing army of 4 ranged/2 melee units for defense.
7. Don't build horsemen/knights. Ever. They suck.
7. Try to have at least 1 coastal city, if you're on continents. The importance of this, besides the fact that fish/sea luxes are good tiles, is that it allows you to build caravels once astronomy is discovered. Besides $15 from CS's and meeting new trade partners, it allows you to do one thing. It enables you to spot the runaway civ and stop him/her in time. In my latest game, I didn't know Russia was runaway until I was notified she entered atomic while I was on industrial. By the time I gathered a force to attack her, it was too late (see
http://forums.civfanatics.com/showthread.php?t=505249)