I recently got a new road bike. A really beginners one (which is what I am). It is a Triban RC120 with disc cable brakes.
I have two questions,
Are there gear-chainring combinations that I cannot expect to work smoothly? Or I should expect all of them to work just as fine?
When does the normal user change the chainring? When I am in the biggest gear in the smaller chainring and I want more, do I have to change to a smaller gear and then to the bigger chainring?
I believe your bike has a double crankset with 50-34 chainrings.
The general advice is to attempt to avoid cross-chaining, i.e. avoid the big ring and biggest 2-3 cogs in the rear, and avoid the small ring and smallest 2-3 cogs in the rear. There is actually some empirical testing that shows that these combinations create higher friction in the drivetrain, and they likely wear the chain and cassette out faster also.
I'm pretty sure that with almost all groupsets, you can expect the chain to actually shift to all the cogs. In the small ring and smallest 2 cogs, the chain rubs against the front derailer cage quite noisily, but the rear derailer does actually make the shift. In my experience, big ring and biggest two cogs are slightly noisier, and you can often trim the front derailer to reduce the noise (that is, the left shifter often can actuate a partial shift which moves the cage slightly, but not enough to go to the small chainring).
Practically speaking, it is fine to leave the chain on the big ring most of the time, and to shift to the small cog when climbing significant hills. I find that at most normal solo riding speeds, a 50t chainring leaves me in the middle cogs on the cassette. Thus, one rubric I use is that if I hear the chain rubbing the front derailer slightly, I'll downshift in the front (i.e. shift to small chainring, which decreases the gear ratio) and simultaneously upshift in the rear (i.e. shift to smaller cogs, which increases the gear ratio). With the small rings on modern setups (i.e. 34t or 36t), many riders will probably be in the smaller rear cogs on flat terrain, which is not optimal for reasons discussed above.
In the past, when typical gearing was 53/39 and cassettes had tighter ranges (e.g. 12-25 or 12-23 in the rear), riders may have been advised to leave the chain on the small ring most of the time. I certainly recall being advised to do this by people in my cycling club. I believe it was conventional wisdom that you wanted to learn to spin the cranks smoothly, at a relatively high cadence. I don't know that there was a lot of empirical basis behind this advice, i.e. it could just have been fashion. Most newer cyclists will pedal at relatively low cadence, and will probably work their way up to a more optimal cadence gradually.
NB: please read what Adam and Argenti said about overlapping gear ratios between your two chainrings.
To answer your second question, you can think of the chainrings as your high range and low range, and the sprockets on your cassette as your position inside the range. These ranges overlap a lot. You'll want to experiment a little (and it will depend on your exact gearing setup), but when you're in (or near) your little-little combination and you want to shift to a harder gear, you'll shift to the big ring and downshift on your cassette by 3 gears (it will be easier to downshift in back first). This calculator helps you visualize what's going on (I've linked to a setup that is very close to yours, but not identical).
As Adam says there is several ratios overlap between the ratios available in the small and large rings, so you can change from the large to small chainring, then 3 sprocket up in the rear and be in approximately the same ratio overall.
It's a good idea to anticipate which chainring you'll want to be in for the section or road or trail coming up. I run in the big ring most of the time. When I see a steep or long hill coming up, as the gradient starts to increase I'll drop to the small ring, quickly change up one or two gears in the rear (which effectively drops down one or two ratios), then continue shifting down in the rear as the gradient increases. I'll do the opposite as I come over the top of the hill to get into the big ring early for the descent the other side.
For what it's worth, the way I ride is I start off in small ring and then change up to big ring once I hit the middle of the cassette. So around the 18/21t mark. Depending on how fast I'm decelerating/accelerating, I sometimes have to change up or down on the cassette to compensate for the front change. That seems to be a 'feature' of riding with a compact.
My older Triban 3 has a triple chainset. On that, I start off in gear 3 on the cassette and middle ring. For me this is fine for starting on a flat gradient. I'll change to big ring once I get to 3rd from smallest cog. Going uphill, I'll go to smallest ring once I get back to 3,2,1 cogs.
I find that the best approach to avoid any chance of cross-chaining. I know modern kit should be able to cope with it but just because it can do it doesn't necessarily mean you should in my opinion.
