Are we dreaming?
Yes: Everything is a dream, ephemeral, temporary, constantly changing, and there is no inherent "existence" that is "radiated" by other "things" in this "world" (time x space) apart from our "own" mind.
No: This is not a dream, this is "real life" and independent of "mind" or "awareness" there will always persist an existence, which is generated by "others" in the environment, and no product of "personal" awareness/intelligence/sentience.
This topic is difficult to address with experimental science because we have affirmed some observer/observee obstacles to overcome in making experiments.
Generally: Science assumes that the human being is an unbiased, observer-on-the-sidelines, basically looking in through a window at "the world" and making talk of its "physics"
However, the view of the "observer" is actually biased, since the observer is necessarily part of the universe that s/he studies and observes. It is like a knot in the thread, trying to untie itself.
How can you understand, truly, if "you" is dreaming, or if "you" is part of a coherent, greater reality?
Well, you should not rule out the simultaneity of yes/no, and the non-possibility of simultaneous yes/no, either. These questions transcend the foundational assumptions of Science and Philosophy, and it is perhaps more intuitive to adopt a "flexible" set of fundamental thoughts, since it seems like we really want to talk about Awareness.
How Aware is one of a dream? You may be engaged in it quite forcefully, invested all your "chips" and have many Possessions, Achievements, Feelings and Thoughts all within this "sand box reality." Then, one wakes up and all understanding of gain, all benefits stemming from self-concern (concern for personal wealth, personal happiness) go poof. Gone.
The "memory" may persist, but in reality there was nothing "real" about the dream. Maybe it hurts to be apart from personal happiness, but really it shows something very interesting: Based on our observation of the situation, our feelings changed. If we could understand the deeper truth of how Dreams/Reality inter-played, perhaps we could understand more about how Reality/"Beyond" might inter-play. Since, if Reality is a dream, there should be something that transcends it in all directions... When one "wakes up", perhaps a similar "poof" may occur to all our cravings and attachments (all our foundations of personal/self-awareness).
We are constantly confronted with the self/non-self duality in daily life: we are a consistent, rhythmically breathing piece of the greater universe, yet we feel separated at large. Really, we observe from a "personal" point of view, but the universe itself is still greater than us all, and certainly contains "us". We are all of the same fabric, and all teammates playing for the same team. Perhaps it is a strange dream...
If it is a dream, how does one wake up? Well, this "awakening" is commonly referred to as "Enlightenment" and it is said to come from a deep inner understanding of the "true nature of things" -- to really understand yourself as a harmonious part of the universe, to not view others as "others", and to really come to grips with the transience/temporary nature of "life". One knows "Enlightenment" like one would know Hot and Cold -- it is perhaps one of the few desires founded on wisdom and it is the desire to "liberate the mind."
How does one get there? What does it mean?
Welcome to the Matrix, here are your introductory pondering questions; try and resist the urge to cover them with thoughts and words (be "silent" in there!), and let the "self" contemplate these things in the tranquility of the peaceful, unobstructed mind:
Everything changes. Nothing is permanent.
Around you, there is infinite creation and infinite destruction in every moment.
Happiness is actually awesome.
"I" strives to be happy and understood. True happiness comes from within, since all originates from the "mind" and this "observational" point of view. Consider that you can reflect negativity or happiness to the environment and those around you. Stuff you reflect gets reflected back eventually.
Continuity of the "one" object.
We say the universe is made of pieces-parts, but really, you could never just delete a planet from the universe and still call it the universe. You'd compromise the "structural integrity" in a way, but really you would be changing a lot in order to make sure one planet got "deleted" because it's all part of one continuous flow, and you'd need energy that was somehow "extra" energy -- energy that was beyond all the types of energy our universe is used to. Something that could both ease the flow and "remove" things...
Divisions of objects/things:
Every "object" that one can identify is just an agreed-upon convention for naming/communicating. Really, at any instant, you can't say that "you" is made of the same atoms, feelings, thoughts, or state of mind. Everything is in flux, so where is "you"? Where is any object? It is founded in the mind, and if you get good at watching/observing your own mind, you can start to understand the "formation" of desires and the "formation" of perception of objects (reification).
Understanding that the observer is still a part of the "observed" is a good place to start in trying to attain the insight you want, I think. Everyone's approach is different, although sometimes there is significant overlap in coming to "true understanding" between people and methods. Really, the process of living is finding out which method is right for you. If you hear something wise, and believe it to be so, do not discredit it due to the source. A dog's bark might enlighten you one day, you never know what your spark, what your trigger may be.
The more you contemplate it, the more you will be able to explain to others your ideas. Really, you should be patient until you feel like you've found the right question to ask. Even if you have access to omniscient wisdom, it's all about asking the right questions.