As discussed in this question certain social activities such as eating could be hard to support between alien species with very different dietary requirements and eating customs. What social activities though would be possible?
We can assume that the aliens in question have biology compatible enough that they can survive in similar conditions to each other.
Only thing we can safely assume of these aliens is that they are sentient and that we have some way of communicating with them. And even then different species might have significant differences in particular areas of mental performance.
Even games of chance usually require estimation of probabilities and reactions to such games would vary between species.
A computerized co-operative game that adjusts gameplay based on the data it has about the players might work. Say a MMORPG that gives each player a character and user interface compatible with their actual abilities, and adjusts all challenges based on the characters in an "additive way" so that all characters are needed and all players involved.
As Pavel Janicek noted a co-op game would help with diplomacy. Additionally players would become more adept at communicating with each other which has value beyond diplomacy. The computer would acquire lots of information about the mental patterns of the players, which would feasibly have value to the species controlling the game, if some players come from cultures that are relatively unknown.
Was going to put this as comment to Pavels answer, but this way I do not need to worry about character count and Tim gets more answers!
Logical games is my best shoot:
And bonus, we can always throw in co-op board game to be played by Romulans, Humans and Klingons together, where the game can be won only by working together thus helping the common diplomacy
Cultural shows, Seminars, Conferences between scientists from different planets about their respective specialties, Tech demonstration CES for computers of the galaxy, trade exhibits on new weapons systems, new starship designs etc. Then there is the annual Nascar Universe
Social activities are derived from shared values. We can see differences here even among human cultures, and with alien cultures it becomes even broader. For example:
Musical and theatrical performances are popular, but some religious men won't watch women perform, some religions and cultures find certain content objectionable, and tastes vary.
Sporting events are popular, but some find the violence in some sports objectionable. As for participating in sports, some value skill and competitiveness and others value inclusiveness.
Social games are popular with some, if there are shared interests. People who favor Trivial Pursuit, Axis & Allies, EuroRails, and Apples to Apples tend to be looking for different things in their participation.
Travel/sight-seeing is something many enjoy, but the "spend the day on the beach" people can clash with the "spend the day in museums" people.
It's entirely possible that a human and an alien will find a common social interest that two humans, or two aliens, wouldn't. So instead of thinking of this at the species/racial level, try to think about the values of individuals or smaller groups.
First and foremost: sex.
Sounds silly? Maybe. But if we assume, that alien species won't have our religious/puritanical views on sexual intercourse it's actually the most logical thing to do.
Most lifeforms we know of on Earth have very strong urge to reproduce at all cost. Among mammals and birds there are studied and documented cases, when a male of one species rapes a member of another (gender of the raped specimen is not always important), because a female of their own is unavailable. I won't provide links for that, google at your own risk.
Taking that into account it is safe to assume, that aliens would be driven by similar instincts - they did evolve somehow, didn't they?
So what would be the point of interspecies mating between aliens? Pleasure, boredom, a desire to produce a hybrid superior to both parents... take your pick.
Now that we have that out of way, let's focus on something less likely to mark my post as a spam.
General information exchange is a good start. That's what most adult humans do, when they meet in groups - smaller or larger. The topics depend on participants. It may be weather, gossip, sports, cars, books, politics, or million other things.
Why wouldn't aliens simply talk to each other? They surely have a lot of information to share, whether it's something about their planets or something about current events.
Once we're done with the small talk, how about business? Different parts of the universe surely have different technology, resources, designs, artworks... It doesn't have to be some big business contract:
"Hey Zog, what a nice necklace you have there, and my Bama would kill me if I won't bring her something nice. I've heard iron is rare on your planet, how about we trade?"
"Sorry, Zama, that belonged to my mother. But I have a different one in my luggage, let me show you"
Feel free to take it from here. As long as there is stuff in this Universe, there is someone willing to buy it and someone willing to sell it.
Some games, physical or intellectual, would surely be in order as long as you have two species valuing the same thing in one room. If math skills are necessary to find a mate for species Leibniz an Newton they will be throwing calculus at each other before you know it.
Some poster mentioned "equalizing chances". Why would anyone even try to come up with such a game? There will always be someone with an advantage and someone with a disadvantage, no matter how you design the game. This holds up even to such a small group as the few of us answering this very question, not to mention entire universe.
I suspect that the rules will be mostly crated ad hoc, and will be somehow based on what is important for survival on different planets. We humans highly regard games like chess where you need to think, because it was our brain, not our muscles which helped us eat the mammoths (on the other hand it's the big muscle guy who pushes the brainiac out of the way and takes the female, but hey, it hasn't been long since we climbed down from those trees).
And lastly - religion. In my first paragraphs I assumed, that it's not an issue, but it's your story - you can have your aliens as religious as you want. And every self-respecting religion tries to find as many followers as it can. So why they shouldn't try to convert each other? That can be a funny story arc to write. Especially if some of those religions have some really strange rituals. In this case - there is nothing limiting your creativity. You can sit back and enjoy yourself.
