I have read many articles which say that amplifying such quartz oscillations should not be easy.
But these articles are wrong; if anything is "easy" to amplify, it's a pure sine (you can filter away all harmonics if your amplifier generates them), as generated by a harmonic quartz oscillator, or the square wave you typically get out of a cleaned-up output of a quartz Pierce oscillator.
So, never mind these articles.
how do the radio amateurs broadcast at stable frequency with power limitations
You buy or build a stable oscillator, for example based on a quartz crystal, you use that and modulate some information-carrying signal onto it (be it AM speech, FSK data or anything else) and then you amplify.
That's harder than just amplifying a pure sine wave as coming from a harmonic oscillator, but frankly, there's 130 years of experience doing that. It's a solved problem, with many different solutions, depending on the frequency, power, bandwidth, dynamic range, power efficiency and signal quality requirements your transmitter has. There's not one "golden circuit" that does everything from 10 kHz to 10 THz :)
Either by coils amplified with standard Class A-F amplifiers or by quartz components which have better stability?
Question makes no sense, coils are components, but you amplify signals, not components. And quartzes and coils are used for different things in a transmitter: not everything you need for a transmitter is used to build an oscillator or an amplifier!
The reality is that if you professionally design a transmitter for an amateur radio-"typical" bandwidth and an amateur radio-"typical" frequency range (meaning: less than 20 MHz bandwidth at carrier frequencies below 8000 MHz), and you don't do it for nostalgic reasons, then you will probably design it such that you use digital signal processing to generate a lower-frequency version of the signals you want to transmit on a carrier frequency, convert that to the analog domain with a Digital-to-Analog Converter (DAC), and then use a commercial tone synthesizer to generate an oscillation that then is used in a mixer, mixing up your lower-frequency version to the frequency you wanted. Then you add an appropriately designed amplifier (again, what kind of amplifier you need there depends on how much power you need, how clean you need it, on which bandwidth, with how much difference between the weakest and strongest output, and at which power consumption).
Both a DAC and a modern frequency synthesizer will need a clock source – these days, you'd typically use quartz oscillator components, or SAW oscillator components, as frequency reference (but there's many more ways, especially if your actual frequency reference comes from outside!). Again, what you choose there depends on your requirements – some communication systems need you to have a frequency that is stable less than 100 parts per billion of frequency error, others will still work when your transmitter is wrong by 1/1000 of its nominal frequency. So, choices to be made really depend on what you need your transmitter to do!
How can I amplify for a higher power rate using simply a quartz?
"power rate" is not a correct term; I think you just mean "power". Please try to be exact with your terminology. Adding "rate" to something can drastically change the meaning! Electronics, just as any kind of engineering, depends on understanding what you do precisely. Having a misunderstanding just because you have abused some words, that would be sad :)
You can't amplify using a quartz. A quartz is not an amplifier.
You can amplify the signal that comes from an oscillator that was built around a quartz crystal, I guess that's what you're asking here. Again, being precise in your head about what does what in a system is very important to understanding a system. That's why we literally teach students who study to learn how to build electronics as the very first thing how to analyze systems made of components, and what the individual components do. "Precisely breaking down functionality into components" is a core skill in understanding technology! It probably is the core skill!
I would need for example 5 Watt power by using a quartz. How can I achieve this?
First of all, you need a license to emit that much power, and even with one, you're not allowed to just emit a constant unmodulated tone with no information content on its own, in most countries. And you need to use one of the frequency bands that you have a license for.
So, learn for your amateur license, because you are not allowed without, and the license test requires you to have learned a few fundamental things about RF electronics. Not much, but a few fundamentals, which would help you.
Then: To technically realize this, you look at the frequency you want to make your quartz oscillate at (again, the quartz itself is just a part of the oscillator – not the signal, and not the whole oscillator), then decide whether you rather
- generate a low-power clean sinusoidal signal using a quartz crystal in an oscillator circuit (e.g., a low amplitude Butler oscillator), and amplify that or
- generate a low-power rectangular signal using a quartz crystal in a different oscillator circuit (e.g., a Pierce oscillator), and amplify that.
The first case makes the oscillator slightly harder to build (arguably), but you don't need to filter as much, as long as you build an amplifier that's linear enough.
The second case makes the oscillator very easy to build (arguably), and the amplifier just becomes a literal pair of uncompensated transistors, so very easy, but you need to filter the high-power signal.
You can buy both kinds of oscillator circuits ready-made, including the quartz crystal, as so-called "quartz oscillator" (note that this is more than just a "quartz crystal" or "quartz resonator"), or build them from components.
You can by both kinds of amplifiers, or build them from components.
What you do, buy or build, is up to you. How you can build, and what you can buy, will mostly depend on the frequency you want to work with, and you haven't mentioned that, so it's impossible to even point you in a definite direction.
Below is a circuit which I have found on internet. This should give only an idea what I mean.
No offense, but that means you haven't really reached the point where you understand enough electronics to understand these circuits – you browse around for them. Nobody needs to understand all the details of every oscillator type they build, but if you're just "shopping around" for oscillator circuits, I'd strongly recommend you just by a complete quartz oscillator component, instead of buying just the quartz crystal and building the oscillator yourself. The commercial oscillators are going to be better than anything you'll build yourself, and that's not your fault! They're better than what any of us could build from discrete components at home, because technologically, it's wise to integrate these things in a small component, and fine-tune everything at the factory with very expensive equipment instead of at home.
