Build offensive units, never defensive ones (unless Riflemen or above if needed for border defense or escorting attacking/artillery stacks)
To add to that, build Veteran units rather than Regulars whenever possible, since Vets are 'stronger' per unit. Similarly (this was pointed out to me recently), building more cheap units is often better than building fewer expensive units (especially if you are resource-limited), because the AI also 'values' quantity over quality. If you haven't already, I would strongly recommend that you also read '
Warmongering 101' for some general advice and '
Inner Workings of the Military Advisor' for the numbers on which that advice was based.
Examples:
2 Warriors (2*10s=20s, A=1+1) are counted as 'stronger' than 1 Archer (20s, A=2), which is itself 'stronger' than 1 Spear (20s, A=1) -- and Warriors can be upgraded to Swords as soon as you have access to Iron, and the cash (60g per unit, before Leo's)
Less intuitively, 3 Chariots (3*20s=60s, A=1+1+1) are not only 'stronger' than 2 Horses (2*30s=60s, A=2+2), you get 3 units for the price of 2, and the Chariot->Horse upgrade costs only 30g per unit, which is a lot less than the Warrior->Sword upgrade. And redlined Chariots/Horses have a 50% chance to retreat (to heal, if you're careful), which Warriors/Swords don't.
So if you can build two high-shield Barracks cities in the early game, get them to 5 SPT and 7 SPT, and pump out Warriors and Archers/Chariots, even if you can't build anything else.
Also, regarding this...
My "stacked deck" consists of 8 traits (incl. commercial and agricultural) and 4 starting techs (Bronze Working, Map Making, Iron Working, and Literature...
So you're not actually playing Conquests, you're playing a mod...
I can win at Monarch without doing any of this, and I'm not the world's best Civ3 player by a long shot -- although there are (or have been) plenty of them on here...
The above 'military strength' calculations also show that you're not really doing yourself any favours by giving yourself IronWorking at the start. If you don't have Iron on your turf, you can't use it anyway. If you do, and you hook it up immediately, you won't be able to build Warriors anymore -- which forces you to build Swords at 30s, instead of 3 times as many Warriors for the same shield cost (and one-third the build time per unit).
Besides, the AI luuuurrrves IronWorking, so you can almost always trade for it, at Monarch and likely also at Emperor (although less reliably at DG+) --
after you've already built a massive Warrior-army to upgrade -- by beelining down the less favoured middle branches of the AA tech-tree and getting a monopoly on e.g. CoL, Philo (NB take those two -- in that order -- for the Republic-slingshot!), Literature or Maps.
And this...
I go after The Great Lighthouse right away, then The Great Library, and later on Leonardo's Workshop (to cut my upgrade costs in half).
Maybe, juuust maybe, your military is weak because you are pouring shields into these Wonders, instead of building units...? As Theov points out, even at Monarch,
every AICiv gets a Settler, a Worker, and 3 mil-units to start with -- you start with only a Settler+Worker, so you're already 'weak' against, well, everyone! Although the AI's production-discount at Monarch is only 10% (and it's very wasteful), the longer you wait to start building mil-units, and the fewer towns you devote to doing so (because you've tied up the best ones with multi-turn megaprojects), the 'weaker' you will get -- and the more (often) you will be forced into paying tribute. Perhaps you should also read
this article too...
But if you insist on having your Wonders, then remember that you don't actually need to wait until you get a tech before you start accumulating shields for them. Instead, you can start 'prebuilds' 10-20T ahead of time, i.e. high-shield projects (e.g. Courthouses, Markets, Aqueducts, the Forbidden Palace, other Wonders) which you can use as 'shield-banks' for later conversion into your chosen Wonder once you get the tech for it -- provided that you haven't 'rushed' any shields into the project with e.g. forest-chops, unit-disbands, whipped citizens, etc.
And BTW, in a coastal city, the Colossus is a better use of Wonder-shields than the GLight...
And this...
I've been building libraries in every town (to enhance research), and after education I've been building universities.
That's fine, so long as your Sci% is also kept high enough for it to be worthwhile.
Just remember that all 'multiplier' buildings only provide their advertised benefits if your Income spending percentages are set accordingly, especially in the early game when most towns are still small (<Pop6), and/or producing <10-15 uncorrupted trade per turn (when the game's rounding-down of the Trade --> Tax.Sci.Lux conversions has a much more noticeable effect).
This article explains it in detail.
So Libs and especially Unis need a decent Sci% (say, ≥20%) to boost a small town's beaker-output noticeably, whereas Markets and Banks only work as tax multipliers if Tax% (= 100% - Sci% - Lux%) is likewise ≥20%.
But again, all the time your cities are building improvements, they aren't building units (and don't forget the maintenance costs)...
Even though I have stacked the deck heavily in my favor, this particular time I just cannot seem to keep up, much less get ahead. 
