What are some of the key things to keep in mind when moving to a higher level of gameplay? What are some tips that I can use to adjust to the higher levels?
Fewer wonders, more units, streamlined infrastructure, more micromanaging, closer attention to diplomacy, and oh yeah, fewer mistakes
Most of this is self-explanatory, regarding streamlining your infrastructure, I found that prince->king was the jump where I began to need to apply my warlord->prince philosophy regarding wonder selection (sure it would help, and I want it, but do I
need it?) to building selection. The capital usually needs all the buildings, and buying them the turn they become available (particularly science buildings from uni's on) is one of the best ways to use gold surpluses. Other cities, not so much. Having just made the move to BNW, this particularly applies to culture buildings: with all culture buildings now only providing 1 culture (with the potential for more if they ever get a great work in there, but let's face it, they won't) and still costing 1 maintenance, monuments are the only cultural buildings needed as they pop borders. gold buildings are great as they are maintenance-free and always add some gold, but with the other priorities that you have, there's often something more beneficial to build until the city is making at least 7GPT (7+1(market) = 8 +25% = 10, hence market = 3GPT, more as it makes more). Growth buildings certainly help, and I'm all about growth, Monty's been my favorite civ since vanilla, but growth buildings are a gamble. There's obvious exeptions to all the above (UB's, Izzy with NW's etc., as well as gold buildings to get yourself out of the red), just use your head and ask yourself whenever you're making something other than a unit or trade unit, "Do I need this?"
Regarding the OP, it sounds like you prefer domination games, and I find that King->Emporer is the level where I needed to play more nasty for dom wins. It's no longer about building a huge army and marching them onto your victim. Now it's about keeping that army mobile and ready, paying attention to when a neighbor gets into a war with a civ on the opposite side from you, waiting 5-8 turns for them to move their army in the opposite direction, and only then storming their lands when there's 1/5 the resistance present. I don't know if you're playing G&K or BNW, but as I'm just adjusting to BNW, I'm finding that I need to turtle a while longer and build up before engaging; maybe I just haven't figured out all the tricks yet. Any advice beyond that would require an overview of your play, there's lots of tricks found here in the forum, like mass building/buying units before their upgrade becomes available and then mass-upgrading. This is particularly effective with archer->Cbows. Can't wait to try out Shaka and insta-upgrade a dozen impi. Getting off track, point is that's one of hundreds of tricks that are found here that will make the transition easier.