You're right about it being expensive. I don't think I'll be much help, since I am of the 'clubbing' variety - I came here with an explicit hate of clubbing, and then spent all of Freshers in the Union and in clubs in town, and now I'm hooked.
If you just want a 'quiet night' with some of your new friends:
#Talybont Social [if you apply to live up here] - Sure, it's a bar, but it's more like a boy's club than anything special. It'll be packed on match days and in Freshers, but after that, it's dead. Pool tables, Ping Pong, some music on the TV, and bartenders who won't look at you stupid if you say "I'll have a pint of Cola please."
#The Taf - if you're closer to town, in say, Senghenydd or Gordon Hall, or even Colum/Aberconway/Aberdare - head to the Union's own pub. I know you say you don't like drinking, but, again, this is one of the "cheap" options at a social night out where you can still say "Pint of Cola" or "Glass of tap water" without the bar staff thinking you're some kind of pussy like they do in town.
#Theatres - you don't have to go every weekend. Look up the shows that are on over the next few months, pick one or two, and book to see them. There's the New Theatre in town, there's the Millenium Centre down in the Bay; heck, you could be 'really cool' and go out to Canton and the Chapter Arts Centre. They hold a Latin American Film Week every year.
#Eat Out - yes, expensive, but it depends on where you go. Avoid eating out on match days; but, if you have a few mates who hate clubbing, decide to go out for a Wetherspoons cheap meal...
#Walks - at least on the sunnier days, a walk around Bute Park or Roath Park never goes amiss. It's really nice, especially walking down from the Talybont end. Go and play Frisbee over in Pontcanna field - take a bike and cycle the Taff Trail. It's all good fun.
Really, as everyone else has said, you've got to let us know your interests before we can suggest things that would be "good" for you to do.
And yes, while there are umpteen societies for you to join, many of them have their 'nights out' where the aim is to get as wasted as possible, it seems. The French Society had 'Drink France Dry', which featured free-flowing wine. I think the only ones that haven't had "Let's all get ******" meetings are the OddSoc, and the Spanish-Italian Society.