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.