The right time to approach people might be before and after class, or during break if there's a break. You don't really need to know names or important topics to start a conversation. Most people like someone who takes an interest in them You mention "small convos" you've had. Next time you meet a kid you talked to before, you can say, "Hi! How's it going with [name the topic you had talked about in that "small convo]?"
That will show them:
- You remember some little thing they said two days ago, which means:
a. you listened when they talked, and
b. you're interested in them.
- You're a friendly person who might turn into a new friend.
Most of the kids probably feel a lot like you do, lost and lonely, not sure how to make new friends. Seeing a friendly person open up in this group of strangers might help the others open up, too.
As for names, you can say with dignity like lots of older people do, "I don't know about you, but I couldn't remember everybody's name that first day. I'm Joe/Jane. And you are?" Possibly, when you offer your name, they'll tell theirs without being asked.
These things are called "breaking the ice." Once the ice is broken and people are feeling that it's okay to be casual and friendly, saying hi to each other even if they can't remember names, conversations will take care of themselves. It will feel awkward at first but breaking ice is hard work by definition. If the first person does not respond well, you can try it with another. Maybe the first person was just having a really bad day. There's sure to be someone in that large batch of kids that you will click with.