I recently made this step up myself, it's actually not that big of a step.
You want to focus on pop growth and science, research for writing and then what improves your luxury resources. Prioritise getting the NC, I find I can usually comfortably get it around turn 80 with at least 2 cities out.
Scout early and build archers. Check the demographics to see who is the military leader, if they're near you, you can expect an attack if your army is too small. On a standard size game I've never been attacked if I'm in the top 3 for military.
I ignore most of the early wonders apart from the temple of artemis which i've never missed when going for it. I recently moved up to emperor and got it it no problem on my first game.
Trade routes are essential for the early sciece boost and use them internally to boost food to your capital to boost growth there. Use tehm to boost a newly formed city too. I strongly favour costal cities as the trade ship bonus is massive compared to land ones. Once I'm tech leader I will tend to switch trade routes to city states rather than other civs.
Research wise after i've got the NC i'll get up to iron working on the bottom then straight to education, fill in the bottom a bit then straight to scientific theory. Make sure you get the newly researched science building built and populated asap, buy them in low production cities if you have to.
After education I'll usually take astronomy and get exploring (i usually play continents) and after sci theory I will head straight for artillery then platstics for research labs then rocketry and satellites. I'll usually have a great engineer ready to rush Hubble as it's a great wonder even if you're not going for a science win as it gives science a huge boost which can help other victory conditions.
In general I'll sell all my luxury resources for GPT or lump sum with allies, i'll trade or buy ones I don't have if I need the happiness. I'll often sell embassies to the AI for GPT early on, it's easy to get an embassy later on, even sell open borders to non agressive civs far away too.
Get city states as allies too, they send you lots of juicy goodies and you can troll the AI with the world congress and more importantly stop the AI shafting you.
Wonders I always go for:
Temple of artemis
Forbidden Palace
Hubble
Nice to haves:
Colossus - extra trade routes. The AI will usually beat me to this as I'm usually late to iron working by prioritising the NC so I don't sually try for it, if no one has baged it after a reasonable time I'll get it
Broubador (sp?) if I found a religion
The one whch I forget that gives you a tonne of happiness, AI really goes for that i find.