I'm trying to learn instrumentation but then I still don't understand the range of these instruments. It's too complicated for me to understand the range, key and notation. Can anyone explain it to me?
There are a few issues here. Let's start with notation:
Many instruments are written in concert pitch. This means that, if you write a C, that instrument will play a C. The (standard) flute and timpani are concert-pitch instruments. Other instruments are pitched in a particular key; hence a B-flat Clarinet and B-flat trumpet, if playing a written C, will actually play a different pitch. There's an easy rule for this:
When playing a written C, the instrument will sound its name.
Thus a B-flat trumpet, if it plays a written C, will actually sound as a B-flat (="sounds its name"). The same is true for the B-flat Clarinet.
(And from here forward I'm assuming the use of B-flat Trumpet and Clarinet, which are the most common "versions" of these instruments. But there are also clarinets and trumpets pitched in C [concert pitch], D, E-flat, A, etc.)
Regarding ranges, I made up this little chart. These things are always subjective, and obviously high school players will have a different range than professional orchestra players, etc., but it's a general idea:
Note that these are the written pitches. The flute and timpani are concert pitch, so these pitches are what you will here. For the Clarinet and Trumpet, these pitches are actually written a whole step (major second) higher than they will sound.
Regarding the timpani, the joined notes indicate common ranges for each of the four drums.
So, a little quiz: You want your trumpet player to play an F-sharp. What pitch should you write? Well, if a written C sounds its name (B-flat), then it sounds a whole step lower. Thus you must notate the pitch a whole step higher. For your trumpet to play an F-sharp, you must notate a G-sharp. The same is true for the clarinet.
Similarly, let's say you're looking at a B-flat Clarinet part and they play a written E major scale. Recall that, when playing a written C, the B-flat Clarinet will sound its name (B-flat); thus it sounds a whole step (or major second) lower. The E major scale thus sounds like a D major scale.
It's a confusing process at first, no doubt; just keep working with it, and it will get easier with time!
(There are also some fun clef tricks you can do to make this all easier, but that's another answer for another time!)