Great question - I remember when I myself was confused about this very same thing many years ago, and indeed at first, it all seems completely random. In order to answer your question, there needs to be a little background:
Historically, thinking about music in terms of harmonic progression is one that has really only come to complete prominence in the last 150 years or so (not very long when considering the past 2000 years...) Mozart was coming out of the Baroque period, and during that time, many people thought of music linearly - as in, vertical harmonies are the result of linear motion, not the other way around. Mahler and Sibelius are a little more difficult to pin-down - Sibelius wrote a lot of programmatic and nature-oriented music, and through his technique of teleological genesis could be argued that in his own way, he was thinking linearly (or perhaps by phrase modules) moreso than harmonically, but I digress.
It is important to remember that in addition to the composers you mentioned, many other composers thought about larger implications of key relationships, and as a result, invented a myriad number of devices to facilitate movement between these relationships. These devices (tonicization, modulation, chromatic saturation) and many others would take too long to describe accurately in a simple answer. But, it is important to remember that they used these devices, and you have not begun to use them. Therefore, your music will not sound like there music.
When writing music, it comes differently for some people in terms of inspiration, and should come differently for people depending on the project. For example, if you're writing a song for your band, you'll probably come up with chord progression first. However, if you're setting text for an art song or working on a sonata for tuba or something, you'll probably come up with a couple of riffs you like before you add chords.
Moral: Don't panic; start with whatever you hear first.
Yes, there are some "canned" progressions. For example:
A lot of composers (especially the famous ones) were regarded for their economic use of chords - they repeated them over and over again. This use provided harmonic consistency throughout the work, and locally, phrases.
You may or may not be familiar with scales and keys. With popular music, often keys are ignored, which can make it challenging if you want to put chords together that sounds like they should go together.
If you are reading the Piston (good text, through incredibly dry) then you should know that chords in major keys can be represented by each degree of the scale like this:
I, ii, iii, IV, V, vi, vii°
In a common progression (I-IV-V-I for example), a chord or chords may be substituted with chords that share like notes. IV may be substituted with ii or vi, I may be substituted with vi or iii, V may be substituted with iii or vii° for example. (It should be noted that iii is an incredibly weak chord and cannot be effectively used as a substitute for V.)
It would be a good idea to learn about the different types of cadences and how / when to use them. You can then use the canned progressions to direct your subordinate harmonies to point toward the cadences and to give your piece forward motion.
Start with a simple canned chord progression and make it an exercise to write some music using only those chords. Do this in all 12 keys. Once you feel comfortable, begin occasionally substituting some of the chords for more interesting harmonic possibilities. You will eventually develop your own style about what you like to hear and how it should be notated.