What bonuses do each of the different city-state types give, and do they ever change?
Military will periodically gift you units. Cultural give you a boost to your per turn culture accumulation. Maritime gives you extra food at your capital, and small bonuses elsewhere.
Which military units you get depends on the age you're in, as does the amount of the bonuses for the other two, so there is a benefit to getting a tech from a new age to increase the bonuses. Being Allied instead of just friendly will also increase the size of the bonus or the frequency with which you are given units.
Its also worth mentioning that regardless of the type of city-state, if you are allied with them, they will give you any luxury or strategic resources that are within their borders. This can be useful to trigger "We love the king" days in your cities to increase the growth rate. When cities demand a resource you don't have, you can use the Diplomacy summary screen to see which city-states have that resource and then consider boosting your rep with them.
Cultural give your empire culture each turn you are at least friendly with them
Maritime gives your cities extra food (favoring the capital)
Militaristic gives your civilization contemporary military units.