You need to provide a little more information about where you are located and which direction your windows face. While the list provided by Linger is good in just about any case, they generally only save minor amounts of energy. If you are in a location that supports it, I would just turn off all heating and cooling. For example, I used to live in an apartment in Boulder, CO with south facing windows that didn't even have AC, and we never turned the heat on, even when the temperature fell well below freezing. This was possible because we opened and closed the shades correctly to take advantage of the Southern exposure, and kept the windos open at night during the Summer to try to "night flush" the air. But this is only applicable to moderate climates.
For very hot and dry climates, getting a smaller portable swamp cooler will save lots of energy.
Hot wet climates are a little unfortunate however (probably a reason humans haven't really developed in these places, since sweating, the same mechanism swamp coolers use, is ineffective).
In really cold climates, the best option is to attempt to take advantage of sunlight when it is out, and when it is not, close all the windows as much as possible with insulating (or just thick cloth) shades.
One interesting way to test your potential for passive heating and cooling is to turn off all heating and cooling for a period of time, say a week or two (unless it gets below freezing often, then have the heating on just enough so you don't freeze pipes...) and try to figure out what works best for your apartment at your location as far as operating shades and windows, and wearing heavier clothes and blankets.
On a side note, you mentioned that you cannot install a programmable thermostat, but in many apartments that have older, simpler thermostats, the wires are actually all there to replace it with a programmable one, so you may want to check. You can always remove it when you leave and the landlord will never know, if they care.