[RPG] Weapon Proficiency without needing a particular class/race or multi-classing

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.

Other than the obvious Feat and multi-classing there is a section on training in the DMG and thus totally optional, but it is on p231.

• The character gains inspiration daily at dawn for 1d4+6 days.
• The character gains proficiency in a skill.
• The character gains a feat.

If you can get a feat by doing this or proficiency in a skill (pointing out that these don't count against your normal feat progression), I don't see anything game breaking to train for a single weapon. This is covered under unusual and usually unavailable training as a reward so it could be something to work toward by gaining renown with a guy that knows a guy for example.

This approach is entirely in the wheelhouse of DM allowance though and I would never allow it out of the box to be honest, as in Level 1.

This answer is based on the assumption that you already have a Wizard created and not necessarily an elf of any flavor, since the question was edited a few times by a couple of moderators this point may not have been clear, as in whether generating one from Level 1 or an existing Wizard of any race.