Generally it is usually fairly straightforward to use an astronomical telescope. Most of the time after arrival at an observatory can be spent checking the instrument and detector performance rather than checking the telescope performance. You should carefully consider, however, how to maximize your efficiency at the telescope; telescope time is expensive and hard to come by.
Before going to a telescope, you might consider the following checklist of things to do:
Remember to consider the calibration observations you will need to take (we'll talk more about this later). Also remember to plan for different possible conditions. For example, if you program requires photometric weather, what will you do if it's not photometric? what if the seeing is horrible, etc.?
One of the first things to be done just after dark is focussing the telescope. This generally involves taking images at a range of focus settings and comparing them to determine the best focus. One should be prepared with software for analyzing image quality to make this determination - also, software for looking at all images simultaneously. Generally, the focus position in encoded somehow so one gets a quantitative measure of the secondary location. One should be aware, however, of the possibility of slack in the gears controlling the focus mechanism, which can make the focus not repeat even when the readout position is the same; because of this, it is generally wise to always move to a focus position from one direction.
While focussing, one can generally also get an idea of the quality of the seeing of the night. Remember that seeing varies from frame to frame, and because of this, multiple exposures even at the same focus can look very different. To minimize seeing effects, one may wish to choose a focus star on which exposures of several seconds can be made: for a brighter star with very short exposures, seeing changes may confuse you. Clearly, however, one doesn't want to use a very faint star because one would like to get the focussing procedure over as quickly as possible so you can get on with your science. Remember, however, the signal-to-noise gains are substantial for a more concentrated image, so it will be worth your while to do a good job: if you rush it, you may regret it later when you have more time to notice how blurred your images are!
You also need to remember that the focus is likely to change throughout the night as the temperature changes. So continue to inpsect your images as you take them, and if the quality appears to be degrading, you should redo a focus run. Most likely, the telescope focus will consistently change in one direction (as it gets colder) and you may even be able to get a good estimate of how much it changes as a function of the temperature with experience. Which direction focus goes is a good thing to write down at a telescope, as you can save significant time during mid-night focus changes if you already know which direction you need to go. (but always beware of someone coming and rewiring the focus motor/control since the time of your last run!).
One may wish to quickly inspect an out-of-focus image for signs of large aberrations in the system. Almost certainly, nothing will be done about these immediately, but if the image quality is poor enough, it may be possible to have something (e.g., alignment) done the next day, so you still may possibily help your observing run, or certainly, you will help subsequent observers. At least, if something seems strange, you should let someone know so they can judge for themselves if there is really a problem.
Overall, this is a key point; you need to be vigilant to look for peculiarities in your data, and if you see something that hasn't previously been documented or that you don't understand, you need to ask someone about it rather than just assume that it is ``normal''!