You can conquer cities in different ways. How you do it depends only on what kinds of cities you've got.
So - if you've got som military cities - just build lots of combat units and send them to destroy the enemy. To capture enemy city you would have to destroy it's buildings and damage the city hall. It takes some time.
If the city is defended by cannons and units - destroy them first, then attack civilian structures. And watch out for citizens attacking you with grenades - unfortunately, you can do nothing about it.
Capturing cities using religion doesn't differs much from military approach. Create propaganda units and send them to enemy city.
You can buy other cities only by using your economic city. To do so - create vehicles and send them to the city you want to buy to create a trade route.
Since then - your vehicles would travel from your city to another, filling the special bar.
When it's full - you can buy the city. It's cost may vary.
You can negotiate it's price - but watch out. If your offer is correct - the city is yours, but it is insufficient - you would have to wait for bar to be refilled once again.
Defending the city
Build about 2-3 defense turrets - they should do the trick. And have some boats in case of water attack - ships have greater attack distanse than the turrets.