I am going to a singles mixer in a few days. I'm an introvert and find
it hard to make small talk with people I don't know.
I think I can console you a bit on that point: Most people find it hard to make small talk with people they don't know - because, honestly, it is harder than with people you know. However, the good news is: Like many things, it's an acquired skill - it's something you can learn (and it's not too hard, IMHO), and the best way to learn is to practise.
How do I make the most of a singles mixer (that is, meet people and
have meaningful conversations)?
First of all, it probably helps to define your own expectations. What do you want?
- Are you looking for new friends?
- Are you looking for people to share a hobby with?
- Are you looking for a life partner?
- Or are you just curious whether you'll meet interesting people?
All of these are ok (though some are more difficult than others), but it's always helpful to be aware of your own goals. After all, you decided to go to this even, you must have had a reason to do so.
My personal advice would be to just go with an open mind, and see who you meet. If you have a more specific goal, that's totally ok - just don't be disappointed if you don't meet the perfect chess partner or the person to marry.
After that, my main advice would be:
Be yourself, but be flexible.
You don't want to pass as someone you are not, so be honest about what you like, and don't try to play something you're not. At the same time, there is no need to push your views into someone's face - listen to what others have to say, and see where you find common ground. And most importantly - enjoy yourself!
Also: Try to balance listening and talking. Both are important, and many people tend to either speak non-stop when they are insecure, or clam up. If you want to get a conversation going, you need both.
To address your specific concerns:
I find it difficult to talk to people I don't know - I'm not nervous
about speaking to them, but I don't really know what to say, so I just
don't do it.
As mentioned above, most people find this difficult, so you are not alone. The good thing is that the others will be in the same situation as you :-). Just try to strike up a conversation. The key thing is: The topic does not matter too much. If someone feels like chatting with you, the will, and if they don't, they won't, so don't stress too much about what to say. In all likelihood you will find someone that you get along well with.
I don't know how to extend conversations. So if someone asks 'how was
your day?', I might say 'it was good,' and leave it at that, which
ends the conversation unless the other person prods for more. I don't
give more detail on my own because I'm not sure the other person
Well, they asked, so they probably do care (and if they asked without meaning it, that is not your problem). Just talk about what you did or experienced that day. It does not need to be deep and meaningful - you're just getting to know each other. The idea is to tell something about yourself so the other person gets to know you. Just to get you started, some things I find nice to talk about are: the weather (cliché, but really works), a small anecdote from your job or family life, something funny that happened to you, something you enjoyed. As long as you keep it short, and listen to the other side, you'll quickly find out what they like to talk about.
It's easier for me to talk to a single lone individual. It would be
near-impossible for me to inject myself into a group of strangers.
Again, that is difficult for most people. One solution is to primarily focus on talking to others who are also alone. Another solution is just to stand next to a group of people, give a friendly nod, and listen. While normally this might be considered intruding into the group, the whole event you go to is about meeting people, so it should usually be ok. By listening you then get a feel for whether these are people you like to talk to (and vice-versa), and what to talk about. Then speak up when there is an opportunity - or leave, if the group does not feel right for you.
I cannot speak naturally with strangers (unlike with family and
friends) and would need to think through what I was planning to say
before saying it. Mostly because I don't know them and I don't know
how they will respond.
Again, this is quite normal. My experience is that this gets better the more you do it. And if it really happens, honestly admitting this is also ok. You may meet someone who feels the same :-).