Phrasal verbs are, by their nature, common. :) Something such as a process (say, a legal trial) "gets underway", but if a person is doing the action, then the pattern is "to get [something] underway".

Yes, the phrasal verb 'get under way' is correct and fairly common. For example, if you say 'The building work for the new school has got under way' it means that it has started. We can see from this and your previous post that you are keen to use phrasal verbs. But please remember that it is important for you to learn correct, grammatical everyday English before you try to use more complex expressions. Phrasal verbs are difficult, because you have to get them exactly right, and if you get them the slightest bit wrong - or try to use them in a sentence which is ungrammatical - people will not understand you. It doesn't matter whether you use expressions like 'get under way' or not - it is fine to say 'start' or 'begin'. I'm guessing that you're a chef, so let me give you an analogy from the kitchen. You wouldn't decorate a birthday cake with chocolate, cream and candles on the top before you've baked the cake, would you? Well, learning a language is the same. You have to do the basic work before you can start on the fancy bits.

Incidentemente, la origen de la frase es sobre navigación de un barco. Literalmente, «under way» significa que la velocidad de el barco a través del mar es suficiente rapido que la timón funciona, y puede dirigir el barco. Incidentally, the origin of the phrase refers to sailing a ship. Literally, to be "under way" means that the ship is moving through the water fast enough for the rudder to work, so the ship can be steered.