Selamat malam, Pak
Lama tak jumpa dengan Bapak.
Your questions are always good and intriguing.
I'll start with the most common word.
(1) Menaiki (naik)
It literally means climb up on something.
i.e. Dia sedang menaiki tangga ( he's climbing on the ladder)
It means ride on something.
i.e. Aku dapat menaiki sepeda motor ( I can ride a motorcycle )
However, it's more common to say “mengendarai”.
(2) Mendaki (daki)
It refers to climb up a mountain or a hill.
i.e. Ayah gemar mendaki gunung ( Father likes mountain climbing)
(3) Menanjak (tanjak)
As an adjective, it means slightly tilted, it usually refers to the road.
i.e. Jalan agak menanjak ( The road is slightly tilted )
As a verb, it usually indicates someone’s career ascends to a higher position.
i.e. Karirnya sedang menanjak ( his career is ascending)
(4) Memuncak (puncak)
It refers to critical or uptight event.
i.e. Kemarahannya makin memuncak ( His anger was getting worse ) (I'm not sure about the English translation)
I crossed my finger hope that the explanation would help.