The Chili Peppers are a necessity. They are one of those bands that have really been able jump through genres of music, more or less, and do so with finesse.
This might be divisive, but I’d say you should start with Stadium Arcadium. As a whole, the album is a fantastic sampling of the sheer variety in their music. The whole album is wonderful, but some of the key tracks to listen to are:
-Stadium Arcadium and Hey (Slower. Sad. Wonderful.)
-Death of a Martian, Wet Sand, Desecration Smile (Somber, semi-slow; they all have big shifts from verse to chorus, something I am a fan of.)
-21st Century, Hump De Bump and Storm in a Teacup (Funky. Shout the lyrics in your car, at passengers or at other drivers)
-Torture Me, Especially in Michigan, and So Much I (fun verses with musical orgasms in place of choruses)
-Hard to Concentrate and She Looks to Me (The former will play at my theoretical wedding, one day, the latter is a likely candidate as well)
-Make You Feel Better and We Believe (These songs have an angst to them. I love it. They also both build and build to cathartic choruses)
I could put every track on here, because the album is just so very solid. After getting to intimately know this this one, I can try to recommend a next step.
Listen to I’m With You last, though. Not because it’s bad record; I actually love it. But because I think you can appreciate the changes so much more when you see what they deviated from to get to that album.
Also, if you like the genre, check out Cage the Elephant’s self-titled album. They’re one of the few bands that I think are comparable in terms of musical style. (The Chili Peppers kinda-sorta made their own genre of music, in my humble opinion.)