Observable Objects:
The following celestial bodies are observable, and with a successful observation, will allow you to predict the future.

The Sun
All 3 moons (Yavash, Katamba, Xibar)
All 47 constellations
All 12 planets

Different celestial bodies let you predict on different skill sets. Also, many are only viewable at certain times of the year, and night. See my Star Chart section for detailed information.

Making the Observation:
Now that you know WHAT you can observe, I will tell you HOW to observe.

The verb usage is, OBSERVE <object name>. For example: Observe Sun.
Occasionally you will be in the room with another character who has a name that starts with the same first letters as what you are trying to observe. In this case, if you simply OBSERVE <object name> you will probably get this line: "You observe <NAME> with fascination." This can be quite embarrassing, and obviously it's not what you intended. To bypass this, and always ensure that you observe heavenly bodies instead of other players, use the verbs OBSERVE <object name> IN SKY. This will always work, and you might want to get in the habit of using this phrase instead. It takes a few more seconds to type, but you avoid embarrassing moments.

You will know that you have made a successful observation of the object when you see this message: "You learned something useful from your observation." What this means is that you have added a skill set to your "observation pool" for use in predicting. This "observation pool" is quite large, at least 17 observations per skill set. (Note: The observation pool may actually increase with level, more testing needs to be done to confirm this though).
There are FIVE skill sets for the purpose of predictions: MAGIC, SURVIVAL, DEFENSE, OFFENSE, and LORE.
The more stored skill set observations you have, the greater your chances for a "powerful" prediction (larger bonus/longer duration). This could entail a LOT time spent observing if you are wanting to fill up all 5 skill sets, since you must wait at least 3 minutes in between successful observations. (17 x 5 = 85 single observations, x 3 minutes = 255 minutes, or over 4 hours).
Luckily, many celestial bodies have the potential to add more than one skill set to your observation pools when you successfully observe them. It is important to know that when you predict the future for a particular skill, all observations of that skill set are used up, putting you back at ZERO stored observations for that skill set. However, stored observations of other skill sets remain untouched and available for later use.

Observation Messages:
There are several messages you can receive when attempting an observation. I will explain them below. When I use the term "successful observation", I mean that you made an observation that added at least one skill set to your observation pool.

"You see nothing regarding the future." - This indicates a  NON-successful observation. Simply try again. 

"You learned something useful from your observation." - This indicates a successful observation, and that 1 or more skill set(s) have been stored in your observation pool. Regardless of any other messaging, skill sets are NOT added to your observation pool unless you see THIS message. You must wait at least 3 minutes (up to 5 minutes) before you can make another successful observation. It is important to note that if you are observing a celestial body that can add more than one skill set (Yavash for example: Magic and Survival), this message does not necessarily mean you added ALL of those skill sets to your observation pool. It merely indicates you managed to add at least one of them. The higher your skills, the better your chances at adding all of the skills sets that a particular celestial body may have.

"You have not pondered your last observation sufficiently.
You are unable to make use of this latest observation." - This indicates that you would have made a successful observation, but you didn't wait for the observation timer to reset. When this happens, no skill sets are added to your observation pool. Remember, you can only make a "successful" observation approximately every 3-5 minutes.

"Although you learn nothing new of the future, you grasp more of its pattern." - This indicates that your observation pool was already full of one of the skill sets that you were trying to add. This prevents you from adding any skill sets to your observation pool, even skill sets you may not currently have stored.

"Too many futures cloud your mind - you learn nothing." - This indicates that your observation pool was already full of ALL the skill sets you were trying to add. Obviously, this prevents you from adding any skill sets to your observation pool.

"Your observations are hindered by the clouds." - You were unable to attempt an observation due to cloud cover. No Round Time generated for this failed attempt.

"Your observations are hindered by the clouds.
You are able, however, to make your observation." - This indicates that there was cloud cover, but you were able to attempt an observation. Remember, this doesn't mean you made a successful observation, it just means you were able to attempt an observation.

Factors that affect your chance of success:
When you first start out as a Moon Mage, observations are somewhat hard to accomplish. This is due to several things. In time, as your skills grow, you will start to more easily make observations. Below is a list of things that (I believe) influence your chance of success, as well as tips on how to increase that chance.

Astrology - With the new prediction system, I believe this is the now the biggest factor.

Perception - Also an important factor in determining your success.

Scholarship - Another importat skill for observing, though to a lesser degree than the first two skills listed.

Cloud Cover - Cloud cover can range from making observing a little harder, to completely impossible. Beware the clouds!

Clear Vision - This spell will give you a bonus to your Perception. This is the best way to increase your chances.

*Aura Sight - This spell may increase your odds at making a better observation, but I don't think so. I think it only effects your prediction results when you actually PREDICT. Not to mention it actually decreases your Perception, which will hinder your chances. Personally I advise that you don't cast Aura Sight until you are ready to make your prediction.

Telescopes - While this doesn't actually increase your Perception, it does increase your chances to make a successful observation, and increases your chances to add multiple skill sets to your observation pool if you are observing a celestial body that has more than one.
To use most telescopes, first make sure it is extended (OPEN, PULL, or EXTEND TELESCOPE), then do the following: CENTER TELESCOPE ON <object name>.
Then you: FOCUS TELESCOPE until you have the telescope focused as well as you can.
Finally you must: PEER TELESCOPE.
At this point you make your observation attempt. If you don't succed, keep PEERing until you do, or you need to CENTER the telescope again. *Note: Injuries can prevent you from using a telescope.

Piercing Gaze - This spell will allow you to penetrate most cloud cover. This spell does not however affect your base chances. It is simply designed to remove the penalty clouds place on making an observation.

Still don't understand? :-) Worry not. I will give an example to illustrate the various points I mentioned above.

We will use the Sun for this example.
We start with 0 skill sets in our observation pool.
First I OBSERVE SUN IN SKY until I make a successful observation.
I know I have made a successful observation when I see "You learned something useful from your observation"
We know that the Sun allows us to predict on the skill sets SURVIVAL and OFFENSE.
Thanks to our successful observation of the Sun, the skill sets Survival and Offense are now in our observation pool awaiting use. These are stored until we use them, or lose them.
We now predict the future for our Offensive skill set. This depletes all Offense skill set observations stored in our observation pool. We only had 1 anyway. :-)
Our observation pool now contains only 1 skill set observation, which is for Hand Survival.

That concludes the section on Making Observations. Now that you know how observations work, you should go read about Predicting The Future.