My party has just completed the second book of the Age of Ashes adventure path and have rescued a young Red Dragon who was supposed to be sacrificed.

Now the party wants to make the Red Dragon an ally and take him back to their Castle (from the previous book), and the Dragon is down for that because living in a Castle sounds pretty neat for building up a hoard of treasure.

So my question is: What kind of shenanigans would a young Red Dragon want to get up to?

I had a group that traveled with a young blue for quite a while, since they had similar goals.  Reds, though, haha...they are some CE beasts.  Sounds like you are open to having the players try to make it work.  

~Plenty of freedom for it to express its chaotic side may reduce the likelihood of that strong evil bent that will be a powerful thing as it gets older.  
~Is it maybe female?  Female evil dragons can get crushes, at least that is what fantasy books say.
~Is it a warzone? Let the dragon do what it does best.  Might be collateral damage now and then, but that is war.
~Greed is possessiveness.  It is going to get really territorial.  A piece of the castle is going to have to be fully treated as its own, or it is going to start either taking over bigger parts of it or dig a monstrous cavern in the thing to make it better.
~Reds are going to like to sear their dinner.  Players can join in on seared buffalo every Sunday!  Sure to shock guests.  Until a guest touches the dragon's favor piece of meat.  That guest and probably a couple of others for good measure won't see Monday.
~Characters don't care about monetary wealth?  Dump it down the dragon chute!  Falling into the red's pile deep in the castle's belly.
~Do dragon's need to learn their spells like everyone else?  Maybe!  That red sure would like some help from the PC magic using character. Just don't end up like Hercules's music teacher.

Thanks for the ideas Kuroth.

I think I will ramp it up slowly, just a leeetle bit evil at a time.

IF you decide yep, it's CE, and always will be, think of your typical abusive relationship; the dragon LOVES the one PC, and that PC IS HIS.  He doesn't WANT to hurt her, but SHE NEEDS TO LEARN.  Do the little things.  First dogs go missing when the PCs pass through town.  Later, small children.  Do it behind their back.

I'm dealing with a black dragon hatchling that the D-born paladin is raising in our campaign.  She's masked (magically) so she looks like a copper, and needs to keep her head down.