[RPG] How to decide which of the intended classes to start as when multiclassing in D&D 5e

Your PC can invent a new technique on his own

Taken from wikipedia's entry on the Southern Praying Mantis kung fu style:

Praying Mantis

The association of the term "Praying Mantis" with the style is also controversial. Each branch of the style offers a different explanation.

The traditions of the Chow Gar and Kwong Sai Jook Lum branches each maintain that their respective founders Chow Ah-Nam and Som Dot created their styles after witnessing a praying mantis fight and defeat a bird. Such inspiration is a recurring motif in the Chinese martial arts and can be found in the legends of Northern Praying Mantis, both White Crane styles, T'ai chi ch'uan, and Wing Chun.

If your rogue is observant enough - he may become inspired by some natural phenomenon, gaining insights about efficient and effective motions useful for offense, defense and maneuvering.

He doesn't need to be an acetic philosopher (though he may grow to become one as he advances) - he just needs the conviction and self-discipline to push him to go through the physical training required to hone his technique.

If your group's play style allows placing more emphasis on narrative at the cost of poorer combat optimization, you can even select to make the transition of your PC's reliance on his new martial skills gradual - continue wearing armor, combine armed attacks with unarmed strikes (not necessarily in the same round, though), etc. But, eventually your PC will start feeling that his armor is limiting him, preventing him from escaping blows that using his technique he can now avoid - then he may consider giving up on wearing armor. Same goes for unarmed strikes - have him hold on to his dagger, but occasionally use a flurry of blows⁽¹⁾ using a fist and an elbow of his other, empty hand - and let him realize that his hands have become faster and more accurate and than any weapon he is used to. This can lead him to relinquish the use of melee weapons, to use an interchanging combination of weapon and unarmed strikes or perhaps even seek "better" weapons which are more compatible with his developing abilities.

Bottom line - just because the "metagame concept" says your PC can now do something, doesn't automatically means he is aware of that and trusts himself enough to give up his old ways completely and immediately.

Or, he can find written records of a technique

Taken from wikipedia's entry on the Northern Praying Mantis kung fu style:

Origins

There are many legends surrounding the creation of Northern Praying Mantis boxing. One legend attributes the creation of Mantis fist to the Song Dynasty when Abbot Fu Ju [...] supposedly invited Wang Lang and seventeen other masters to come and improve the martial arts of Shaolin.
The Abbot recorded all of the techniques in a manual called the Mishou ("Secret Hands")
[...]
This manual supposedly disappeared until the Qianlong reign era,
[...]
The manual records Wang Lang "absorbed and equalized all previous techniques" learned from the 17 other masters.

Aside from self mastering a technique, your rogue may find a documentation of a certain technique - this can come instead of having him invent it from scratch, or as a latter supplement giving him more inspiration and advanced abilities.

At either case, you can probably work the narrative details with your GM so they won't clash terribly with the campaign settings (assuming that having a monk in it doesn't do that in the first place).

(1) Or whatever equivalent term used in 5e for that...

You choose only one of the options. This isn't actually ambiguous in English—choosing from a lettered list is a convention that always means "or" except when specifically stated that more than one can be chosen.

The ses in the words like arms or defences do not indicate anything to the contrary — that's another feature of English that's potentially confusing, but also unambiguous. In English, the s can mean either a word that's grammatically plural or grammatically categorical. Categoricals are words that indicate represent kind without meaning singular or plural. You would not say “choose your defence” or “choose your arm” — at least, not without being grammatically incorrect. For those uses, the categorical must be used, and the categorical is formed the same as the plural — but does not mean plural.

So the Druid must choose either Hide Armour or a Wooden Shield but not both, and similarly for the other classes with these choices.