Is it correct to say ‘I picked up a guy on the street’ or ‘I picked up a guy from the street’?
What’s the difference between pick up something on and pick up something from?
Those are both possible, and it’s hard to identify clear-cut differences in meaning. I think both could be used to describe a police arrest. If you’re talking about meeting someone for a casual sexual encounter then possibly the first is more likely. There may be other differences in certain special contexts that don’t immediately come to mind.
“pick up” has lots of idiomatic meanings (en.wiktionary.org/wiki/pick_up lists 18!). Their usages and applicable prepositions are highly idiomatic and hard to explain without just compiling a huge list. “pick up X on Y” can be used when “on Y” is idiomatic in itself, typically describing the location in which the picking up happens. This is the case with your example, where “on the street” is the pertinent expression. You could also pick up something “on the beach”, “on the island”, “on the way home”, etc.
You may also find “on” used to describe that the item being picked up is physically resting on something else. For example, “pick up the vase on the table” means that the vase is currently lying on the table. “pick up the vase from the table” ultimately describes the same action, but emphasises more that picking up the vase lifts it from the surface of the table.