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?
If someone shares a story and you have a similar story to share, that is usually a good entry point. Don't do this everytime someone shares a story - the goal should be to make them feel like you heard them and can relate in some way rather than to actually share.
Another good way to show that you are actively participating in the conversation is to ask follow-up questions. If they are talking about a recent vacation, for example, you can ask them what made them choose the destination they ended up at. Most of the time you don't actually care about this information, but that's the price you pay to engage. If you're clever about it, however, you can sometimes use these follow-up questions to steer the conversation toward a topic you're actually interested in.
Note that neurotypicals will call you manipulative if they catch you steering things toward your special interests too often. They're right, I guess, but they all do it all the time, too - they just don't notice because they don't have to put conscious effort into it. If people resist your topic, just try your best to set it aside for another time.