It has always been my impression that conservatives are more concerned with individual rights (right to bear arms, for example)

This particular issue doesn't actually support your point. You have an impression of the argument but you aren't actually sympathizing with, or understanding, your opponent. Conservatives, its true, might be said to be worried about how many guns they, personally, own. But so far as the issue itself, conservatives really do believe that the more guns out there the better. They would sooner see everyone armed than only a few armed. This relates to the second half of that article of the bill of the rights and the purpose for which it was written. We have a right to bear arms and some might suggest a requirement to form well-regulated militias. These militias would be local groups, under a local leader, with local interests. Their purpose was to act as a check on the power of the federal government.

To the mind of a conservative, the federal government is a necessary evil. Local leadership is the best leadership and the scope of any federal government ought to be limited. Now, where we go from first causes to today's muddied down, inhereted ideas, that can seem pointless. And yet, to the OPs point, it is still valid.

Justice and peace are important conservative values. To the conservative mind, justice is swifter when delivered by the aggrieved and peace is more likely assured where everyone is on their best behavior (because everyone else is capable of delivering swift justice). This seems harsh to the liberal mind and many conservatives might even disagree with my characterization but the underlying values-motives are still true and speak directly to what the OP had to skim over.

Liberals trust courts and fear vigilante mobs. Liberals trust 'the system' and authority. Not to say that conservatives all distrust the courts or prefer vigilante mobs, but to say they are more focused on the ideals those are supposed to deliver and are quicker to note when they fail (and willing to deliver in other ways). This is also not to say that conservatives are the party or group of anti-authority as liberals often to seem to have a monopoly on that expression. It's all to say that it is complicated and that these issues are much easier to appreciate if you are intentional about a sympathetic stance. Keep asking 'why' until you understand and you'll find it a lot easier to find the middle ground that you desire. Why do you want to own guns? Why is swift justice more comforting that assured justice? Why do you fear a meddling federal government? And so on.

If you learn to ask questions in a generous spirit you will find that the person you are interrogating is fully understandable and all too human.

while liberals are more concerned with betterment of the society as a whole.

This is rather unfair. It is entirely based upon characterizations. Would you think any conservative anywhere would proudly proclaim they are only concerned with their own betterment or that their political aims properly achieved would not lead to a better society?

Closer to the truth, and the way conservatives actually see it, is that liberals pick and choose their battles around certain disadvantaged minorities and are not so much concerned with policies that affect everyone. Welfare is available to all? True, but only used by a few. LGBT rights, only advanced on behalf of the LGBT community (whether that leads to a better society for all is kinda the point of the debate). In the abortion debate, liberals are concerned with this specific woman and her rights while conservatives are concerned with the affect upon the entire society if this practice becomes widespread. And so on.

To further this, if we look at the extremes of the right and left, we have socialists on the left, who significantly favor societal needs over individual needs, and libertarians on the right, who are mostly concerned with personal freedom.

This seems fair at first glance but again we have to look at these in light of the values that were laid out in the post you responded to. It is perfectly fair to say that a socialist society is not very caring. We set one standard for all to assure that none falls too low. The liberal mind can't stand to think of a single person in poverty. In that great leveling, I prefer the conservative values that allow for the exceptional. The notion of fairness is noble but at the expense of the great?

We can certainly come at it from the other direction as well. Libertarians seem to be ruggedly individualistic but I believe that is a means and not an end. That cult of the individual is a reaction to socialism as much as it is a movement of its own. At least here, it grew out of the very democratic ideals that the country was founded upon. We didn't used to say that all men are equal. They clearly are not. What we tried to guarantee was that all men had equal opportunity. Our democracy was actually a meritocracy, a reaction against aristocracy. It was repugnant to our minds that someone should inherit power, that someone should exercise power if they were unfit to it. So, we elevate people based upon their merits. Whether born low or high, all are equal in the sense that all are free to rise or sink to the level they, individually, are capable of. Libertarianism is the harshest example of that. They get rid of the state that can't help those who struggle to help themselves and just let everyone succeed or fail as their personal merits allow.

Of course, to most of us, neither pure libertarianism nor pure socialism are ideal or even suitable. They are useful only to define the extremes between which we should be seeking to find today's balance. The values that motivate them, however, are useful to investigate so far as they shed light on the values of those on the right or left, their nearest neighbors. I believe that, though it is difficult, when you investigate those you will find they are consistent with what /u/themysteriousdoctorx was talking about. After all, he was quoting some real science.