I'd imagine that talking to smart people who don't subscribe to the belief in question would be more productive than trying to dig for inner hurt.

This article was absolutely essential in my rejecting Christianity. There had been thoughts quickly darting in and out of my head for years that I never let myself think fully: "Heaven sounds boring and I don't want to go there." "What the Bible says about killing nonbelievers is abhorrent." What the Bible says about killing rape victims is even worse." They were quick, emotional responses. Little, "Hmm, that doesn't seem right" moments that I always tucked away in short order with a "God is all-wise, stop questioning," etc. I'm not sure why, but I actually took this article's simple piece of advice: focus on what's actually painful to focus on. There's a series of dark-side strategies involved with guiding people to do exactly the opposite, "God is all wise, you are stupid" being a small part of it.

When you're doubting one of your most cherished beliefs, close your eyes, empty your mind, grit your teeth, and deliberately think about whatever hurts the most. Don't rehearse standard objections whose standard counters would make you feel better. Ask yourself what smart people who disagree would say to your first reply, and your second reply. Whenever you catch yourself flinching away from an objection you fleetingly thought of, drag it out into the forefront of your mind. Punch yourself in the solar plexus. Stick a knife in your heart, and wiggle to widen the hole.

There are lots of things that make this especially difficult for certain sorts of beliefs. Here are a couple:

1. The smart objections to your belief may be difficult to find. There may actually be lots of stupid but popular objections to your belief; so in order to find the smart objections, you may have to do a good deal of unpleasant research deep behind enemy lines. If your belief is commonly ridiculed, and those who object to it typically denounce you as a scoundrel for even entertaining it, then good luck finding smart objections to it before you become too pissed-off to reason very successfully about them.
2. The smart objections to your belief may be written in a vocabulary you're unfamiliar with. For instance, if your belief is about political economy (socialism or libertarianism, let's say), you may actually need to get into some pretty serious economics to understand the smart objections. It is easy to state naïve beliefs about political economy whose best rebuttal is a technical one; sound-bite objections such as Bastiat's parable of the broken window, or Keynes' "markets can remain irrational a lot longer than you and I can remain solvent" are probably not the smartest objections.

I don't think this is a matter of training, but a matter of instinct. People don't think about the real weak points of their beliefs for the same reason they don't touch an oven's red-hot burners; it's painful.

You let go of red-hot burners because of instinct. You stop touching them because the pain trains you not to. This is a matter of training. It's just training that it's impossible to not undergo.