About Republic: The general idea is:
Republic Commerce:
- All citizens must be working roaded tiles. Pretty good so far.
- Focus on growing your cities to size 7+, as they earn more gold and unit support.
- Get the luxuries. You have 3 hooked up

- Once you have all of the above, you're economy is pretty much stabilized.
- Don't forget to build marketplaces (helpful when you have 3+ luxuries) and libraries.
- Don't be afraid to use the luxury slider.
- Just remember 50% science in Republic = 100% science in monarchy.
Republic Military:
- Eventually disband outdated units, they hurt support too much.
- At lower difficulties, almost all your units should be offense and workers (see below)
Republic Warmongering:
As long as you don't war for 100-turns-at-once, war wearyness can be managed, and your economy will still be better than Monarchy.
1) Peace
2) Pick a target and declare war (make alliances only as needed)
2) Blitz with all offensive units. You were making all-offense, right?
3) Take a few cities. Destroy silly AI counterattacks with offensive units.
4) Make peace for techs, gold, etc... (don't trash your reputation)
5) War weariness goes away,
6) Go back to step 1
The only thing bad about republic is you need to watch your cities, and adjust them as their happiness adjusts. You should always be doing this anyway, but some people just don't wanna do it.
I suggest:
a) Press F1 key
b) 'At-a-glance' scroll down the cities list and make sure the group on the left is the bigger / same size as the group on the right.
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Republic spearmen
This whole process is why spears are bad. There's no need for them in the blitz -> settle -> blitz -> settle process.
And the best defense is scouting & lots of attack units. That kills anything anywhere. Not spears.
Literally, your spears just sit in your cities, eating gold and unit support.
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About Wealth:
Wealth is rarely useful. It does have limited uses, but it's very rare.
For the most part, in the early game, think of this:
City-Wealth: 20 turns = 20 extra gold
City-Units: 20 turns = 3 extra swords infantry
Basically, most builds are better than wealth (ie units / workers / settlers).
You almost always need more units. If you have too many then go expand !
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About warring in general.
- At lower levels you don't really need catapults. Just mass offense (see above)
- Catapults are good for slower wars, but with republic it's blitz -> peace -> blitz -> etc...
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About diplomacy:
Have you been using diplomacy?
- Demand peace from enemies. Beat all their money / techs / cities out of them

- Trade for any techs you don't have.

- Use alliances to make world-wars work to your advantage.
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About knights and MD Infantry
There is some debate as to whether their worth the shields.
70Shields = 1 knight = 2 move attack unit
40Shields = 1 MDI = 1 move attack unit
They both have their ups-and-downs.