I've been trying to make social connections, but the problem is that any time I start to think about how I can start conversations, the conditions are never ideal. Mainly, everyone is either doing a combination of being busy with some activity or talking to other people. How do I join in on the group conversation? Especially if I don't know most/all of the people there?
This is a difficult problem for basically anyone, so it helps to remember that nearly everyone struggles with this in unfamiliar environments. Even people who don't struggle with this problem probably had to develop the skill by just trying and failing. It takes practice.
A way to approach it, assuming that you're in an environment where it's expected that people will mingle, is to listen to conversations until you hear something that resonates with you. If there's a moment to interject, say something positive in response. Just agree with what's being said. People like being validated, and people will like you if you validate them. If you're warmly received, ask a question that gives someone an excuse to elaborate or agree further. Just be nice, smile, and tell people that you like what they're saying, then just sit back and listen. If you feel like you're not getting good feedback, just exit and go talk to someone else. The idea is to test the waters, leave a positive impression, and don't make anyone feel like you're butting in on their conversation or overstaying your welcome.
The trick is to have fun doing it. If you look like you're having fun and helping other people enjoy themselves, people will respond positively and want to include you in the conversation. Worst case scenario, they won't hold it against you for being nice. Practice it and do what feels good, and stop the moment you get overwhelmed or feel uncomfortable.
Lastly, remember that no one will hold it against you for politely trying to make conversation and stepping away if you get the sense that they don't want to chat with you. What people dislike is when others are negative, argumentative, or linger in a conversation when its clear there isn't any reciprocation. Taking little moments to make comments or ask questions and then moving on is a good way to get used to entering conversations naturally. If someone does take what you're saying the wrong way, simply apologize, say you didn't mean anything by it, and move on. Do it often enough and you'll get a sense for when people want to engage further. It takes practice, but it doesn't have to be a big deal.