Invest ranks in Linguistics
Any creature that invests ranks in linguistics gains one language of their choice per rank. While animals are normally barred from taking ranks in linguistics, improved familiars are not animals-- and even if they were their high intelligence score (i.e. >=3) would allow them to invest in and make use of Linguistics except when playing with this anti-animal FAQ post.
Any improved familiar can take the Extra Traits feat. This allows that creature two traits, many of which grant, in addition to other abilities, the ability to speak a given language. For example, the Unintentional Linguist trait can grant the ability to speak Protean. Of special note is the Gesture Expertise trait, which lets you communicate with any creature via gestures and/or sign language, even if they don't know the language you are using, via a DC 15 Linguistics check.
The 2nd-level spell codespeak lets a group of creatures gain the ability to speak, read, write, and understand a new language for 10 minutes/level at a time. No one else except the recipients of the spell (you+1 target/2 levels) can speak the language, though, so you'll need to have Caster Level 4 before this works.
The 1st-level spell Speak Local Language may work better and at a lower level, depending on how your GM interprets the effects of the tongues spell.
The 3rd-level spell (2nd level for partial-casters) Voluminous Vocabulary grants a creature the ability to speak, read, write, and understand a language for 8 hours and explicitly addresses the spell being cast on non-sentient beings. A caster with an optimized Linguistics skill can grant many, many more languages with a single casting of this spell.
Use Magic Items
A Gold Nodule Ioun Stone is a slotless wondrous item costing 5,000 gp that grants the ability to speak, read, and write one language. As a slotless item, any creature can equip it, though the result is unspecified if the bearer lacks a head.
A Ring of Eloquence costs 3,500 gp and is better, but it takes the ring slot.
Otyugh is an olfactory language with a small dictionary published in Pathfinder Chronicles: Dungeon Denizens Revisited. While the limited number of concepts can be constraining, it's good enough to cover a fairly wide set of utilitarian uses. Rather than learning Otyugh mechanically, you can merely instruct your familiar to produce the correct sort of smell when it wants to communicate each idea and then rely on those smells rather than any sort of check to tell what's up. Note that this is too metagame-y without buying the language for many tables.