My mum got a Kindle Fire, and it seems kind of buggy (though maybe that's to do with my mum operating it!), but the iPlayer app doesn't seem to work, the internet connection seems a bit patchy (even if other devices have no problem) though it's much better after installing the Dolphin browser in place of the Silk one which comes with it (why it needs it's own browser and can't run normal Android apps is one of the main reasons I don't like it), the Facebook app on it has trouble updating properly...and it just generally feels awkward to do anything with it.
I definitely think a straight android tablet with the Kindle app is a better plan if you actually want to use the tablet for anything other than reading. Nexus 7 definitely a better bet for covering all uses.