I figured it out by myself! Who would have guessed. If somebody else stumbles onto this with a similar question here's what I ended up doing:
I didn't realize this, but "user" in this case is another JSONObject. Just as you have to define the JSONObject "tweets" you also have to define a JSONObject "user", that comes from within "tweets". You do this like so:
JSONObject user = tweets.getJSONObject("user");
From there you can get access to anything within user by simply using user.getString("x");