It depends on what type of clinometer you have, but it is a fairly simple affair.
The most common type used for geology is a Brunton Pocket Transit, which has been duplicated and imitated by manufacturers such as Leitz, and some cheaper Chinese versions like CST/berger (which are often so poorly made they have incorrect calibration but geology departments like them because they are cheap). They all have a small leveling bubble on one side of the compass which can be adjusted by a lever on the back of the compass. You place the flat side of the transit down against the rock to be measured (so it sits on its side) and adjust the level bubble to be in the center between the two marks. At this point you may want to check if you have selected the greatest dip angle by slowly turning the compass (still on its side with the flat side against the rock) to see if the bubble moves upward. "Upward" would be updip and if you are facing the compass (for a Brunton at least) "upward" would be to your left. If you turn the compass slightly away from the dip and your original setting was correct (being the maximum dip) the bubble will move to your right turning it both ways away from maximum dip angle. This is much easier to see if you are measuring a very steep dip, and when you try to measure a very small dip angle like 2 or 3 degrees it takes much more patience and some trial and error.
Here is an instruction manual:
http://www.kooters.com/pdf/BruntonTransit-inst.pdf
If you are using a clinometer such as a Suunto like this:
http://www.karstsports.com/15027.html
it is actually even easier. The dial in one of these instruments floats in a fluid and so you put it on its side, with the dial visible and facing you, and pivot the instrument around its top end slowly until you reach a maximum reading. If you pivot it further, passing the maximum, the reading will decrease and you will then know you have passed maximum. This maximum will be the true dip. A Stratum pocket transit works much the same as the Suunto clinometer.
With whatever system you are using, all you have to keep in mind is that the maximum dip reading you obtain from a specific place on an outcrop will be the measurement you want. Any time you are not orienting your instrument in the direction of the true dip, you will get a lessor reading on your instrument. Its also good to look around and be sure you are measuring dip on a good place of outcrop, not on some piece of "float" or any area that is not "in place."
If you are ever involved in an exercise where more than one instrument is being used (like a team with more than one compass) you might impress your teacher by testing the calibration of the instruments on a solid surface like a wall (with no metal in it). If the compasses don't give the same reading (which the Chinese compasses often do not) then using the two compasses together will never result in a connected loop, due to the variance in their readings. Dip readings are less likely to be out of calibration but it is worth testing if you have multiple instruments (and you care).
It occured to me that your question might actually be how to measure strike with the compass which is a little more tricky. Here you have to use the other bubble (the round one) and the side of your compass. You want to get the compass level in all directions, so the bubble is in the very center of the round bubble. Now you can take a reading from the compass needle. You have to constantly keep in mind that strike is not unidirectional, and some regions of the world the practice is to record dip and dip direction instead of strike and dip. If you use a compass with quadrants instead of 360 degrees, you should use only the north half of the circle- and always pay close attention to make sure you are not recording inconsistent meaurements. To test yourself, take the compass, take a strike reading, and then turn the compass around and take it again. Note the difference (180 degrees). I measure strike with dip to my right as long as I am facing in the direction of strike. This is known as the right hand rule. Here is an explanation by Earl McBride and Mark Helper from the University of Texas:
http://www.geo.utexas.edu/courses/420k/PDF_files/Brunton_Compass_09.pdf
I hope all this helps. The best way to learn this is to practice, and if you can afford it buying your own pocket transit will greatly accelerate that process. Likewise, building stereograph diagrams in the field while you are taking measurements will not only increase your understanding of how these diagrams represent reality, but will help keep you from messing up your measurements in the field.
http://maps.unomaha.edu/maher/geol3300/lab/lab5.html
Measuring cleavage direction is simply the same process as strike, but instead you are measuring the cleavage planes of the rock.