The Spirit of the Waters elven alternate racial trait gives proficiency in longsword, trident, and net.
Also the Huntmaster cavalier archetype.
More generally for exotic weapon proficiency, the half-elf Ancestral Arms alternate racial trait gives you a free exotic weapon proficiency for which you could choose net.
There's other more long and painful ways - like a dwarven oracle favored class option lets you reduce the nonproficiency penalty by 1 each time you level, so by level 4 you could be net-proficient. Kitsune and Hobgoblins have an option like that too.
There's a bunch of third party classes that give net proficiency, not sure if you just want Paizo-published or not but search in the classes on the d20pfsrd if you want those.