Not including 3.5-only ways of reducing ASF (and boy there's a lot of those), here are the vanilla Pathfinder options for giving you a lower Arcane Spell Failure chance:
Not really a gish, but bards can cast spells in light armor and with shields without incurring ASF.
The Bloodrager is a mix between Sorcerers and Barbarians. They use something similar to the Barbarian's rage, but can also cast spells during it. Bloodragers also ignore arcane spell failure chance in light and medium armor. Because they're a hybrid class, Bloodragers have slow spell advancement and only gain up to 4th level spells. They also don't have spells until level 4, meaning this isn't a great option for a new character.
Spend a swift action to reduce ASF by 10% for the rest of the round. When combined with other reductions, this one can be fairly useful as long as you have your Swift action available. If you're making a gish, the prerequisites are fairly easy to hit as long as you keep your CL up. Later on, this upgrades to Arcane Armor Mastery, which grants a 20% reduction.
The mythic version of this feat provides this bonus passively, without requiring an action to use.
Still Spell removes the somatic components from a spell, increasing its spell level by 1. If you can afford this, it negates ASF entirely, because ASF does not apply to spells without somatic components. This is a good option for spontaneous casters.
Mithral can be fairly expensive. It's 1,000 gp for light armor, 4,000 for medium, 9,000 for heavy, and 1,000 for shields. It reduces the ASF of the armor by 10%, as well as other benefits from being lighter such as a higher DEX bonus and lower ACP. Mithral items are all masterwork and the masterwork price is included in the price of mithral.
Ring of Spell Storing (magic item: 18,000gp minor, 50,000gp regular)
While it doesn't apply to all spells, a spell stored in a Ring of Spell Storing ignores any ASF you might have. Not a terrible solution if you can't reduce your ASF as far as you'd like.
Not really a way to reduce existing spell failure, but a good way to knock off any ASF caused by using a heavy shield. Summons a wall of force shaped like a shield that grants +2 AC and can be turned on and off at will for free. It's a wall of force so it'll also protect against incoporeal attacks, which is always nice. I highly recommend it for gishes.