Here’s my answer- don’t worry about “How to Achieve True Global Unity On Planet Earth”, but rather work on fixing small problems that are achievable. Help the hungry in your community, help clean up what’s dirty, heal the sick, etc. The world is a far, far less violent place than at any time in the past- in general, things are getting better. I think the best way to do your part in helping the world to continue to improve is to be a decent person, help others, be involved in your community, fight bigotry, etc.
Unity? Easy. All I have to do is kill all the rest of you.
Not so easy when you tell us beforehand. 
I think Genesis tells us all we need to know about global unity. One person was pronounced very good. Two people… well, they fell pretty quick. By the time you get to four people, you have your first murder.
In fact, even the OP’s own example:
Find me any family (let along a big family) that demonstrates unity. If we can’t do it as a single nuclear family, how do we do it as a global one?
Stop blowing up other people’s shit?
We’ve known the answer for years. We just haven’t implemented it.
But this is the problem. We aren’t all “one.” This is a nice little sentiment to give people warm fuzzy feelings, but it just isn’t true.
I don’t mean this in a racist or sexist way. I’m not saying that one group is better than another, just that there are real differences among people (and Hell, many of these differences are something that should be celebrated and encouraged, not waved away). The differences exist between groups of all sizes and descriptions, and are based on a multitude of factors. Then, when we throw the differences between individuals within the same groups into the mix, things get really fuzzy.
We are all one species, but we are not all the same.
It seems to me that the rise of mass communications (1000 channel 24-hour news & opinion TV, or internet message boards) has polarised folks, not brought them together.
IMO, most people seem to seek like-minded people and material to bolster the beliefs and biases they already hold, and NOT to have their beliefs challenged.
So… a benevolent dictatorship?
Quis custodiet ipsos custodes?
Who cleans up after the janitors?
That could work, I suppose, but it’s not what I had in mind. Democracies are much less prone to war than autocracies or anocracies are, so for the best result, the world government should be a democracy.
The people of the Western democracies are closer to the unity the OP is on about than anyone: they kill one another the least, they maintain the rule of law, and uphold Enlightenment values of liberty and equality. Because all humans are individuals with individual desires, hopes, ambitions, and wants, unifying everyone behind any ideology besides “Leave other people alone and do your own thing” is doomed to failure, and even that is pretty shaky. Given that the aforementioned Western democracies best embody that ideology, they should be the template.
Really? Back in the 60s, we thought that long before now, Marijuana would be perfectly legal and accepted by society at least on the level of alcoholic beverages. It’s taking longer than we thought.
To address your larger idea, there are two major hurdles, and they are related. We’ve apparently evolved from hairy, murderous creatures who were jealous, controlling, and prone to gang up on their weaker peers. Fast forward a million years or so, and the most clever and most ruthless ape-people have devised a multi-faceted form of ruling their inferiors: Religion, Authoritarianism (government), and Banking. Parse that any way you like, the “have-a-bloody-lots” control the “I think I’m autonomous, except I’m subject to the authority of these guys and their army/ police who are making me do things I think are unreasonable” members of the tribe.
As long as the ruthless, wealthy cavemen control (at the point of a rifle, courtroom, jail) and attempt (with varying degrees of success) to brainwash, with media propoganda, the less powerful and afluent cavemen, there will be no world unity, unless you consider “unity” to be a World Totalitarian State.
Every generation we get slightly better at it. Travel gets cheaper, so people mix more with other cultures. Education and child rearing gets better, so people aren’t violent by default. The world is becoming more peaceful and democratic little-by-little each year.
It’s a slow process.
What makes the OP think people of different nations or ideologies make war because they don’t “understand” each other?
Al Quaeda understands the West just fine- that’s precisely WHY they hate us!
Not the least of which is the fact that when you eat, I’m still hungry, and when you rest, I’m still tired.
Easy, get rid of all the people.
Challenge accepted!
We get to carry each other, carry each other.
No we’re not. We maintain a thin veneer of rationality atop our irrational monkey brains and our instinctual lizard brains. You’ll get much further in understanding the human condition if you start from the standpoint of us as irrational, superstitious creatures than as enlightened beings.