First, info about higher levels. The AI gets free units and unit support as the level gets harder. This is how it is with the DyP mod, but I don't think they altered that (or if they did it's minor):
Chieftain, Warlord, Regent - no free units, no free support
Monarch - 2 free defensive units, 1 free offensive, 4 free unit supports, 1 extra free support per city
Emperor - 4 free def units, 2 free off, one extra free worker, 8 free unit support, 2 more free unit support per city
Deity - 8 free def units, 4 free off units, 1 free settler, 2 free workers, 16 free unit supports, 4 extra per city
As you can easily see, attacking early in high difficulty modes is suicide. There is NO WAY you can catch up immediately with the AI's free units, you're going to need TIME. Only exception is if you get a very very good UU very early on, then you can attempt to swamp one or two neighbours by producing a lot of those units and attacking as soon as you have enough to take even one city (and keep on producing more of course). In those cases, building settlers is secondary until you've cleared up the space, which is harder than it sounds.
Otherwise, don't bother with escorts right at the beginning, unless there are barbarians nearby (try to maneuver the AIs into engaging the barbs if you can. They have a lot more troops than you).
Lick the AIs' collective boots and give them whatever they want so that they don't attack you. They demand your dearly researched tech for free? Quell that pride, and realise that if you don't give it you'll probably lose a couple of cities, not to mention that it will delay you in your build-up by a few dozen turns while you gear your entire empire towards surviving that early defensive war. Giving them the tech is cheaper, you'll get them back later.
On deity, you're still going to need an early war to catch up, but it's useless to attack them with 5 archers. Once you have half a dozen cities or so, start them all into mass production of some fast offensive unit, pick the weakest neighbour, size him down to half his size, then sue for peace and collect free tech. Have maybe one city building settlers to fill up the gaps, and don't bother building your cities too spaced out. Compact is good, you can disband useless clutter cities later, right now they give you good production and easy defence.
Move on to the next weakest neighbour and repeat. Keep that up until you feel you're close to tech parity and you've occupied more land than most AI's (One third to half of a continent should do as a minimum, with a 'continent' being one entire linked area of land. Eg there are 2 to 3 continents on standard sized maps, normally). If you feel you can bully some more stuff out of the AIs, do it. Don't let those armies sit idle, you should always be capturing territory and tech if you can.
There will come a point when your neighbours are too weak to be given tech even by the other AIs, or maybe they don't have contact with other more advanced AIs. At this point, clear them out of any area which you think would be good to settle in, possibly wipe out some of them. Then disband most of your army and switch to democracy or republic or a similar govt.
This ends your initial, expansion phase. Now comes the improvement phase, where you build up that massive empire, get rid of clutter cities progressively, and start researching (and trading) tech aggressively. Don't bother buying tech unless you really need it. Trade tech for tech or tech for lots of gold. Beeline for techs more advanced than what the computer has, then trade those to them for all the techs you skipped and potentially lots of money too. Eventually you'll take the tech lead. Keep on licking the AI's boots if you feel that a war with them could set your build-up back noticeably.
Oh, and if some idiotic civ does declare war on you anyway, make sure you get allies, lots of them. I still remember a huge map game where the russians, on another continent, tried to bully some tech out of me, and I immediately allied with everyone on my continent who could actually be a risk to me (I paid them to ally with me). Result? The other people on my continent dragged in the Russians' neighbours on our side, and about 30 turns later the russians were wiped out by the rampaging french, and I didn't even have to divert any production to build military units.
Anyway, hope all this helps... Good luck.
Daniel