Game i'm currently playing I wanted to get better at early wars as I never got used to expanding through taking city's when there is area to just settle instead.
I have just started war against that Jap fella, year is 550 BC and i'm Casear. I want to use my UU before he has access to his, by waring I do two main things:
Stop him expanding.
Hurt his research (and mine to).
Bit of specifics about this game:
I'm on Prince (think thats it, whatever the first level is where AI gets advantages).
I built 3 city's (Capital + 3 = 4 total), two are going to be nice money makers, two are good production.
I got Iron in my capital area (Prats! yey).
Spammed Prats (5) and a 2 axemen for protection with a couple of chops (half of capital has forests).
Took the city the Jap's put that bordered my captial with no loss. Healed while moving an fresh built Archer down and saved to think about what to do next when I play tommorow with the next city in the "path" being the capital on a hill with archers...
I think previously I tried to expand to much (6 cities) and bring big armies that took to long to make then deploy giving the target time to research and expand. I kept producing prats and an axeman in two of the cities (throwing in an archer) to replace any lost units while they heal in the captured city.
Keep your number of cities low (but good quality), get 5 or so offensive units then go

while building replacements and defensive units, you can cottage your commerce cities while attacking to expand. I would of started the war sooner if I could of grabbed a worker (thats 5-10 turns of hammers that you save to build offensive units) but didn't get the chance.
Hope this helps.