Ask and ye shall receive! While you were typing your response, I was typing up this almost-definitely-more-verbose-than-necessary elaboration on city specialization. How coincidentally fitting ...
It can be tough; in this version, economy and military strength are definitely two separate categories that require their own attentions (as it appears you are already aware) - in previous versions, you could just focus on production [of military units] and your economy generally came along for the ride.
With that in mind, it pays to have some cities focus entirely on economic endeavors. You will wind up hearing a lot here about city specialization, and it is indeed a very good idea in this version. The idea is that of the three primary 'resources' (food, production, commerce), a given city can focus primarily only on one individual one, and therefore be much more efficient at bringing in that one individual resource type. Typically, the most efficient strategy regarding great people is to only have one single GPP (great person points) farm, and as such you will try to have one solitary city in your empire that has an obscene amount of food (2-4+ food bonus tiles and/or multiple flood plains, and primarily farm-type improvements), to support as many specialists as possible to generate your great people faster.
The remainder of your empire should be comprised of a mix of production-focus cities (for military production and the rare wonder), and probably even more economic-focus cities, to maintain your research rate and gold income. Over time you will get a feel for good balances between the two. Production cities should have a lot of high-hammer tile types (strategic resources, grassland hills), and (this is important) enough food to comfortably work all those high-hammer tiles. As such, it is often very nice to have a food bonus or two for the very best production centers, and barring that, the less impressive sites will need the occasional farms to feed all those mines. Economic centers should just be cottage-spammed essentially, with perhaps a farm or two (or food bonus or two) to augment growth, but [almost] the entire remainder of the workable tiles should likely be cottages (as such, large swaths of grassland make great commerce cities, especially when rivers are present. Look to previously jungled areas to provide the king commerce cities in most games). Pure production centers can ignore almost all of the economic buildings and just build granaries, courthouses, barracks, and stables, and crank out troops, whereas economic cities should rarely have to bother with building a barracks or stable or the like. This will add to your efficiency as well, since attempting to build all buildings in all cities will cost you many turns of inefficiency. Oh and btw, note that your really focused economic centers will have very little production, so using the Slavery civic and whipping citizens is the typical way that they will produce most of their buildings.
A decent balance between the two primary city types (production and commerce) should yield a fairly robust empire that won't fall terribly behind in either department (military/economy). If you find yourself consistently falling far behind in one category or the other, you may need to focus more on the respective city type (e.g. if your economy is constantly lagging, try upping the economic cities/production cities ratio). Of course, some cities won't make perfect one-or-the-other type locations, but even the hybrids can focus in one direction or the other.
Finally, note that neither category has an absolute utopic level of achievement - you will never have 'enough' military or economic power, you are simply striving to achieve the most that is reasonably possible as quickly as possible. You'll always want more troops and wish your economy was slightly more powerful. In this regard, don't worry too terribly much about just barely 'keeping up' techwise - on the upper levels, it's the best you can do, and given reasonable troop numbers, you should be fine so long as they are not terribly outdated. Researching techs as the leader costs the most anyway - the more foreign civs know it already, the more bonus you get towards researching it, so being selective about when and where you try for miniscule tech leads can pay off.
Here is a link to an article that goes more in-depth about city specialization, in case it could help. I suppose at warlord you don't need to get too into this kind of thing, but it can be a good thing to be aware of so you can tailor your habits towards it in anticipation of eventually moving up in difficulties.
Hope that may have helped out a bit. Good luck, and welcome to CFC!
