I understand religion makes people happy/grants culture.
Does it do anything else or is that it?
Oh, it does far, far more than that. Well, not 'much more' than that', but it does that soooooo well that it's pretty incredible.
Religion in a city gets you 1 happy and 1 culture as long as it's your State Religion or you have NO State Religion. Late game, with Free Religion, it'll generate another happy.
Having the religion will allow you to build Temples (with the right Tech) which generate an additional 1 happy and 1 culture. Also Monastaries (with the right Tech) which generate 2 culture and +10% research. And with enough Temples built (and the right Tech) you can build the Cathedral level buildings with generate +50% culture, +1 Happiness with Insence, and (if it's the State Religion) another +2 Happiness.
The State Religion in a city coupled with Organized Religion gives a +25% production boost to all buildings and Wonders. It's about as good as a Forge, and it can be gotten earlier than Metal Casting.
The State Religion in a city coupled with Theocracy will give +2 experience to all military units built there. That generally means an additional promotion to newly built units.
So let's see.... That makes an early and mid game potential of +4 or +5 Happy, +4 culture with a +50% multiplier on top, +10% research, and (+25% production or a combat promotion ).
And that's for a general city. Holy Cities and Shrines generate benefits on top of that. Needless to say, religion is quite powerful.
The only downside is that to get the maximum benefit, you have to name a State Religion, which will upset all the civs who have a State Religion different than yours.
With NO State Religion, the best you can get is +2 or +3 Happy, +4 culture with a +50% multiplier on top, and +10% research. That's still very good.
Having a religion that is not your State Religion, you can still get +1 or +2 Happy, +3 culture with a +50% multiplier on top, and a +10% research. Even that's quite good.