That's a good question. I take it you might be referring to "давайте познакомимся"? Of course, we would say "Nice to meet you." almost always after giving our names. But, after that, there are many options. We unfortunately do not have a standard phrase. I think one common method is to ask a question about how the person came to the place you are both in. Another one is to rely on a someone whom you both know and ask them how they know that person. Often we proceed this with "So..." For example: - So, what brings you here? - So, how do you know ...? (... = the person who is hosting an event) - So, I see that... (then ask a question) I hope these ideas help! In summary, the best thing to do is usually ask a question straight away because it would indeed be a little awkard to say "Let's get to know each other".

You could make a nice comment about that person and start a conversation from there. Maybe you like their clothes or their hairstyle, so you could say "hey, your shirt looks really cool" or "I like your style". This way you can already break the ice and the person will feel comfortable from the start.

The question actually sounds bad in English. Saying "would you mind" indicates that you anticipate that getting to know each other would be unpleasant. Here are some suggested replacements: "I've enjoyed meeting you. Perhaps we could meet again sometime and chat some more." "I hope we can meet again sometime. I'd like to hear more about ... (some interest that you have in common)"