Most of us either barely remember our dreams or don't recall dreaming at all, but some experience fantastical dreams that make their normal reality seem like a hazy dream. I think everyone should be more aware of their dreams because it is more than most people believe it is. The fact that we dream alone is an interesting point, one that many people will write off as a biological "blowing off" of the day's steam, or some other unimportant aspect of our body. If you want to learn more about what dreams are made of, there are exercises to help you develop your ability to recall dreams, and much much more. You might be asking "why would I want to remember my dreams"? It isn't that dream recall is what's important. It is the fact that you are able to utilize your consciousness in your dream that is important. Without developing this ability, you cannot recall your dreams readily (which the dream experience alone can provide much insight and information) and you are holding back an important aspect of yourself.
In order to take advantage of the dream state, the first thing you must do is learn to remember your dreams. This is done through some very simple exercises. These helps develop your ability not only to recall your dreams, but to help start making you consciously aware of the inner world while dreaming. The practice is important because it helps develop your inner senses. If you consider that a lot of people do not remember their dreams, it is because their inner senses are not fully operating enough to transfer their dream experience to the brain for memory recall and storage. Those who are more practiced or experienced in dream recall are able to expand their consciousness using their inner senses, some to the point of having full conscious control. I'm not talking about the morning fantasies where you're directing an imaginary situation. I'm talking about the ability to think critically and interact within the dream environment as if it was physically real. For those who are able to reach this level of consciousness while dreaming, have a much wider range of choices for development.
If you're just looking to understand your dreams, that's a personal endeavor. No one else can interpret your dreams except yourself because all experience is subjective. Before you can interpret your dreams, however, you must first realize that everything in your dreams is a symbolic representation of a thought or emotion. They are the dream manifestation of your thoughts and feelings. The life we live in the human body is a physical manifestation of the thoughts and feeling of our self at an "unconscious" (non-physical) level, but so is our "dream" world the thoughts and feelings of our self as viewed from the "conscious" (physical) level. From this side of the sleeping state (the waking state), we view the dream world as fantasy, but from the dreaming state, we view the physical world as a fantasy. In either case, all forms are symbols that represent thought or emotion. If you don't recognize what a dream symbol means to you, consider what that symbol represents emotionally.
The exercises are very simple and obvious, but it requires you to practice using your will. The first thing is to practice focusing. When you are focusing on something, your mental attention is directed towards the sensory data. It doesn't necessarily mean physically directing the body's attention, but rather your mind. An example of this is the difference between looking directly at something and recognizing what's in your view, and focusing on the peripheral information without looking directly at it. If focusing was strictly a physical behavior, you would have to look directly at it to focus, but since it is our minds that actually do the focusing, we do not have to look at something directly to recognize objects. This mental focusing away from the focus of physical attention is what is called "detuning" or "shifting" the consciousness. It is what you do when you daydream, where you are gazing in a direction, but not being aware of what's physically visible in your eye, and instead your focus is what's coming in from your mind, your inner eye.
Focusing (and not concentrating) doesn't mean a straining type, where you're energy is so concentrated that you're exerting too much energy. Focusing means you're attention to detail is sharp. Your natural observational abilities are being used. It means you're attentive to the subject of focus to the exclusion of all other distractions. If your mind tends to wander and jump around from thought to thought, it will be difficult to control your inner senses. The reason why you are unable to focus is because your body tends to control where your senses take your consciousness rather than your will. If you do remember your dreams, you'll notice that most all of the dream experiences are from an observer point of view. You are watching what you're doing in the scene, rather than being in control of yourself. You might find yourself jumping from one situation to another as your mind wanders. These "uncontrolled" dreams from the perspective of your daily ego are the results of your ego consciousness not participating with the inner consciousness. This exercise helps to "awaken" the ego consciousness within the dream reality (psychic development is the opposite, where you awaken the inner consciousness to the physical reality so that the ego and the inner consciousnesses work together. In either case, being able to focus is important). In order to take control of your body with your will, you must develop a habit of stopping the uncontrolled redirection of focus. This can include practicing focusing on a single object for longer and longer periods of time, or focused meditation where you hold a thought for as long as possible. This will develop your ability to focus and be fully aware. Why is it important? Have you ever been on a long drive and just blank stare the surrounding, not noticing any details so that you don't remember anything? This is the same thing that occurs in the dream world without your full conscious focus. You just aren't paying attention to details and so you forget the dream (in the dream world, your inner consciousness is so used to the nature of that reality that it pays no attention to details or things our ego would consider "wrong", just as our ego consciousness does not pay attention to the common things in this reality we consider "natural"). In the inner world, or the non-physical world, all action is thought and all thought is action. There is no "thinking" as we know it in the body. You just do. If you are unable to control your thoughts and actions (which is thought) while in the body, you will be unable to control your experience in the inner world of pure consciousness. Thus, what you recall, if at all possible, would be a jumble of unrecognizable experiences. So the first exercise to train yourself for dream adventures is to control your thoughts and actions through conscious will of focus. Until this first step is developed to some degree, you will have a difficult time remembering your adventures. Here's a website that might be helpful in training for focusing: Concentration Exercises.
The next exercise is changing beliefs. This is probably the most difficult aspect of the training. I'm not talking about beliefs such as your religious point of view or things like that. I'm talking about believing you can remember your dreams and that you are able to have conscious control of it. When you dream, the inner consciousness is active while the ego consciousness is "asleep". In order to wake up that ego consciousness, your inner consciousness must wake it up. This is done through suggestions to the inner consciousness before you go to sleep. Otherwise, the belief system you hold will direct and drive the inner consciousness. It does what you want it to do because it is you. By making simple, affirmative suggestions, you are beginning to change your inner beliefs. But you cannot have opposing beliefs either since they will cancel each other out. So when you do the suggestions, you must try to suspend your disbelief so that you can begin to reprogram your subconscious thoughts. The suggestions should use affirmative wording such as "I remember and recall my dreams with full clarity. I take my consciousness with me in my dreams." Try to avoid sentences that have no definiteness to it, such as "will" or "should" since they do not reinforce absoluteness. This exercise is a type of self-hypnosis, where you are training your subconscious to accept a different belief. It will require patience and diligence, no different from any other exercise. Repetition and practice is all that is required for all of these exercises.
Once you start to remember your dreams, you should get in the habit of recording them right as you awaken so you do not lose the details, which will fade very rapidly. The purpose of this is to train yourself to remember your dreams through concentration of the ego with the physical brain. The act of recording your dreams is to create a reinforcing pattern for your ego consciousness so that it will accept the dream world's experiences and not reject or ignore them. If you do not accept the dream experience, then your inner conscious will follow your beliefs and your dreams will not be remembered. Ultimately, the goal is to be able to "wake up" within your dreams in order to experience the other dimensions of reality. Through suggestions, you can take your ego consciousness with you, with full critical thinking, movements, and willful manipulation. But without the practice of recording your dreams, it may be difficult to remember any of your fully conscious activities. If you do remember, it will simply seem like a dream rather than a real experience, though parts of its vividness may linger in your thoughts.
The last exercise (though I'm sure there are others that can help) is probably the most important. The key to any mental experience is relaxation. If you stress out about making something happen, it tends to tense up the mind with "what if's" and "why not's". Just play this as a fun game, rather than a deep longing or desire. When you're anxious for something, your ego consciousness is too wound up and focused on those thoughts, thus staying the center of attention, such that your inner consciousness is unable to come through which is where the dream experiences begins. Remember, we're trying to awaken the ego consciousness within the inner consciousness' reality. If the ego consciousness cannot fall asleep to "let go" of physical reality, it will have difficulty in accepting the inner reality.
Finally, if you encounter any frightening or undesirable experience within the dream state (or even in physical life), you must try to control your fear through will and take control of your situation. Fear response is what happens when you give up willful control of yourself and allow subconscious reflexes to act. One way of removing the source of fear is to will it away. Take away your focus and it loses its source of energy to remain and draw upon (thought is energy). Most of the time, these experiences are the manifestations of your own inner fears. But remember that you are in control. When you have conscious awareness, you can wake yourself up at any moment. But overcoming your fears is one area of development you should learn if you want to explore the inner reality to any degree. Otherwise your own "demons" and fears will prevent you from continuing with the exploration. You can always call upon helpers and guides for assistance. If one does not seem to respond, it is most likely due to a condition you are creating where you are unable to perceive them. Just remember, thought is action. Whatever conscious or unconscious thoughts you hold at the moment, it becomes reality instantly in the dream world. This is why the focus exercise is helpful in controlling your thoughts through your will.
If you want to know if there is more to reality than just the physical world, I think exploring the dream world is a good first step in answering those questions for yourself. If nothing else, these exercises will also be helpful for your life in this reality.