Wow, I wish I had a 10-inch Schmidt Cassegrain when I was in high school! A couple of things come to mind.
If you have a set up for photography, why not search for asteroids? If you photograph the sky from a dark location over several nights, and if you are looking in the right spot, you may see a faint 'star' shift its position. Looking in the ecliptic plane, this is likely to be an asteroid. You should start by first trying to observe bright asteroids like Ceres. Look at issues of Sky and Telescope or Astronomy magazine to find out where and when to look for a bright asteroid. Then, consider using your camera/telescope to patrol other parts of the ecliptic plane over a period of days or weeks to search for other asteroids.
The most puzzling astronomical phenomena are the gamma ray bursts. For decades, astronomers have seen these bright flashes about once a day coming from nearly every corner of the sky. We don't know what they are because they happen randomly and no one manages to be looking at the right spot at the right time to see what might have changed in the sky before and after the flash. You might take some wide field photographs of the same spot in the sky, and cross your fingers that over several months of photographic surveying every night, that you catch one of these as it happens. You will have to get a list of detected events from one of the folks at the Compton Gamma Ray Observatory's BATSE experiment. See my hotlist for a link to their web site. Explain what you are thinking of doing and the local webmaster might provide guidance. You might not capture a burst, but a collection of photographs of the same part of the sky ( the same constellation) spanning several months could turn up other interesting things too.
You might consider following the light curve of some unusual variable stars as they change their brightness over the course of hours or days. See issues of Sky and Telescope and Astronomy for more info on popular variable stars. You might also consider monitoring the light from very young stars such as T Tauri. There are dozens of others as well that ought to be visible through a 10 inch telescope. These stars are only a few 100 million years old are still settling down to become 'main sequence' stars. They have lots of gas and dust orbiting them, and their variability is related to their very active chromospheres.
Although there is no real science to it, you might consider assembling a photographic atlas of selected star clusters, nebulae and galaxies. You might investigate how the sizes and shapes of nebulae and star clusters change in different parts of the Milky Way.
Counting stars has been used for a century to estimate distances to various astronomical objects. Photograph a piece of sky towards a nebula or dark cloud, and a similar sized piece of sky away from the cloud but nearby to it in the sky. Count all the stars in a reference area from each photograph, keeping track of how many stars you counted per square degree, at each magnitude class. If your counting is deep enough, you will see the effect of the presence of the cloud as a sharp decrease in the star counts at some magnitude level. Knowing the location of this object relative to the Milky Way will tell you from a set of tables, how many stars you should see at each magnitude, and from this you can estimate how far away the cloud is. There are some technical details in making this work that you have to explore.
If you want to study solar system objects, you can always monitor surface features on Jupiter and Mars. This will depend on just how good your telescope is and the quality of your local atmospheric seeing. Don't bother if the stars are twinkling a lot!
Try your hand at searching for comets. Your likelihood of success is very small though! You can study the changes in the appearance of already discovered comets such as the new and exciting 'Hale-Bopp' comet due to arrive in 1998. Photographic studies are best.
The rotation axis of the Earth does one complete cycle every 25,725 years. In 100 years, the location of the north celestial pole moves 5500 seconds of arc relative to the stars. This means that in one month, the axis of the Earth relative to polaris shifts by about 5500/(100*12) = 4.5 seconds of arc. This is a very small amount, but can you think of some way, using photographs taken of the north celestial pole region several months apart, or detecting this slight shift? A several hour time exposure would show the stars trailed into arcs, whose geometric center is the North Celestial Pole. You would now have to have some landmark on the Earth that you could see in the star trail photographs and against which you would measure the pole shift. Very distant street lights perhaps?
This is the best I could do in 15 minutes. If you want to use real astronomical data instead of data you generate from your own telescope there are LOTS of neat things to look into. If you are interested, drop me another line and I will provide some representative suggestions.