Minus the "just walk up to part" lol, during a light conversation (ugh, small talk) I'd change the topic by saying something like, "man, I've got nothin to do after school. If you're free, you wanna hangout?"
I've never done that before because I never initiate turning aquaintances into possible friends... but it would be easy if I desired to. My advice for you (things I also tell myself): Don't worry, overthink, overanalyze, or whatever. Keep it simple, allow yourself to be direct if needed but don't force the social interaction if you're uncomfortable. Relax.
That's why I adamantly omit the "just walk up to part" as I know damn well that would be a possible disaster/failed socialization on my part (ie: crushing awkward pause instead of me changing the topic, due to nervousness or trying to prevent any sensory overload). To reiterate, I would never just walk up to someone to start a conversation as that significantly increases the difficultly.
In case of emergency, a well placed introspective "hmmm..." plus the rubbing of my chin to disarm the awkwardness, giving myself a chance to relax then continue socialization. Disaster/fail would be me mentally retreating, completely, (specifically to deal with nervousness or sensory overload), thus allowing the conversation to fade... unless they continue talking to me, which I may then recover.
Thankfully not too many people look into that or take it personal (as it is never clear what's going on in my head from their perspective).