"If you really want to see for yourself what the Buddha was talking about, you don't need to bother with books. Watch your own mind." - Ajahn Chah, Thai Monk of the Theravada tradition and cofounder of the Chithurst Monastery in Sussex, England.
It is this advice that aspiring Buddhists find to be both greatly difficult and highly rewarding. One can fill themselves with immense scholarly knowledge of Buddhism while experiencing none of its insights. On the other hand, one may know nothing of the Dharma or the Buddha yet know firsthand the wisdom that they teach.
The best way I can suggest of immediately experiencing this is to simply close your eyes and follow your breath - from its initial entrance into your mouth or nose, down into the depths of your lungs, and slowly back out. You may want to internally count each exhalation and see if you can get to 10 without any non-related thoughts arising. If you can, congratulations! You already have developed a calm mind, fertile ground for awareness to bear its fruit. If you are like most people, however, you completed a few full breaths, followed a side-thought off somewhere, then realized your err and resumed counting (this may have happened a couple times en route).
To experience one of the more powerful insights associated with Buddhism, some degree of practice is most likely necessary. Wildmind.org provides a wonderful structure for such practice.
However, as mentioned above, meditation itself is not entirely necessary for one to develop insights into the Truths of existence. While meditation is the practice prescribed by the Buddha to his disciples as a way of developing awareness, such an awareness can be cultivated in simple daily life. It is awareness and concentration that unlock the doors of wisdom in the Buddhist tradition; if one is skilled in these they will grow to understand the purpose of such things as meditation, rituals, scholarship, and the likes.
Too often our minds are too clouded with frivolous, repeated thoughts to operate at our highest capacity. Yet each of us has experienced a time when our minds felt truly lucid and expansive. For many this comes most easily at times of artistic expression, such as painting or writing poetry. This is how I imagine the enlightened mind to be at all times. But while the artist's lucidity is capable of being interupted mid-stream, the enlightened mind is imperturbable. Just as a great painter's brush can manipulate a canvas and a poet may expand a reader's mind, the enlightened individual's impact is without substance, without direction, and without bounds.