In my city I organized movie nights inside of the university and showed Austrian movies with english subtitles (e.g. Wolf Haas movies or "Muttertag") so foreign exchange students could learn about Austria and Austrian humour. There was a welcome party at the beginning of every semester in a big club with 600+ people and a goodbye party at the end of every semester. In between there were movie nights, museum vists, skiing trips, wine road trips, city tours, pub crawls and lots of other stuff to do and get to know Graz and Austria.
All of this is organized by volunteers (ESN members) who do this in their free time. There are city chapters of ESN (ESN Graz, ESN Vienna etc.) and above them a country chapter (ESN Austria). ESN Austria is a platform where all the volunteers meet and exchange information and ideas and talk about what works and what doesn't for our students. ESN Austria organized a skiing trip every year where they booked whole hotel and hundreds of Erasmus students from all over Austria could go there and try out skiing for a few days. Participating was cheap for the Erasmus students because ESN people got sponsors on board (e.g. free Red Bull and lots of other stuff). ESN Austria also had a cooperation with an Austrian telco and if you were an Erasmus students and you come to our office during our office hours, we give you a free sim card and a cheap bank account. The telco and the bank would pay us a few € for every activated sim card and every opened bank account and since ESN is a non-profit that money would go right back to the students (e.g. subsidized museum trips and other cultural events). It's really a win-win-win situation for everyone.
I was never on Erasmus but I was a volunteer for ESN both in my hometown and in our national chapter and due to this I got to know lots of amazing people, both volunteers and students, learned a lot, had a lot of fund and acquired a few skills that would turn out to be quite useful in my later life.