Gunslinger 1/Spellslinger 19 seems like your best bet. Gunslinger 1 gets you Grit, most notably the Quick Clear option early on. Also, getting Gunsmith from both Gunslinger and Spellslinger arguably gets you two battered guns, which is nice with the dual-gun option on Arcane Gun. The single-gun option doesn’t seem worth it, particularly not when you have two guns anyway.
Feats are a problem; you want a lot of them. Human as your race probably makes the most sense. Point-blank Shot, Precise Shot, and Rapid Reload are stupid feat taxes, but you still have to take them. Two-weapon Fighting seems kind of obviously desirable, but I’m not actually convinced that it is: you need a lot of feats, and you will not be Full Attacking often, between the need to reload and the use of Standard Actions to shoot.
Too bad guns suck
The other issue is that, sadly, firearms pretty much suck. They’re expensive (or time consuming) to arm, they are prone to misfire and breaking (especially Arcane Guns), their damage is decent to start with but doesn’t scale very well since you’re not adding any ability scores to it, and they take ages to reload. Rapid Reload is a necessary feat and still isn’t nearly good enough.
School of the Gun is absurd
And then we come to the huge problem: School of the Gun forces you to ban four spell schools. See my Q&A on banning schools; it’s 3.5-based, but ultimately the choice is based on which spells each has, and the spells have not changed all that much between 3.5 and Pathfinder. I recommend banning Enchantment and Evocation to start with, and then Abjuration and Divination so long as you have a cleric who can cover those. If not... I really would not be doing this. Necromancy’s next up, probably still along with Divination even if you don’t have a Cleric; you don’t want to leave your party without dispel magic. You really do not want to give up Conjuration, Illusion, or Transmutation ever.
Frankly, you’re getting nothing but problems here. The gun’s damage isn’t even remotely enough to justify their own misfires and reloading problems, much less the loss of four schools. You don’t even get the Arcane Archer’s trick of throwing Personal-range effects “over there.”
Arcane Archer option
A much, much better approach is to just go Wizard 10/Arcane Archer 10, and call your light crossbow a gun. Maybe talk with your DM and see if you can take firearm-specific feats with the crossbow. But you can keep the crossbow behaving as a crossbow; no need to use firearms’ terrible mechanics. With this, you can also take the fairly-solid Focused Shot feat after Precise Shot, to add Int to your damage.
Straight Gunslinger; take magical feats, traits, and ammunition
The other option is to go straight Gunslinger, and ignore spells entirely. This guide is excellent for that. Notably, Gunsmithing and a rank of Craft (alchemy) can create alchemical cartridges for guns. Further, the Mysterious Stranger archetype is Charisma-based, and that can lead you to having decent skills with Use Magic Device even cross-class (it is a very, very bad sign that cross-class UMD is a serious suggestion, though). Mysterious Strangers can also put the Dangerously Curious and Magical Talent magical traits to good use. Ultimate Magic’s Eldritch Heritage feats come highly recommended, again especially as a Mysterious Stranger (the aforementioned guide goes further and recommends a halfling mysterious stranger, but I’m not sure why). Master Craftsman and Arcane Talent can also allow you to craft true magical ammunition.
Gun Magus homebrew
Finally, this homebrewer has created a Gun Magus archetype that might be worth considering, if you’re allowed homebrew. The ability to “shoot spells” is still quite limited, again falling quite short of what the Arcane Archer can do, but you get grit, deeds, and spells. Furthermore, ErrantX is one of the best d20 System designers out there right now: his homebrew has long been widely regarded as some of the best out there, his long-running homebrew PrC competitions have been wildly popular and have produced some incredible work, and he currently is a professional RPG author for Dreamscarred Press, hired as the Lead Developer of their forthcoming Path of War supplement. If you’re unfamiliar, Dreamscarred Press is responsible for Pathfinder’s psionics system, and are, themselves, extremely well thought of. I would say they know Pathfinder better than Paizo does. They certainly have superior design skills.