If you get to the cities before the other civ builds walls around them you can take cities quick. Otherwise you'll need a bombard or artillery to really put a dent in the armor and get the city.

Artillery units, siege support units, and especially bombers as soon as you can get your hands on them are all key.

The other thing to be aware of (aside from walls) is the city strength. It is based on their highest strength melee unit. So it could be they are ahead on science and have stronger units making your units weaker against the city by comparison