That highly depends if you learn only Kanji or if you learn Japanese (so Kanji + Vocabulary)
For Example nearly every kun-reading of most Kanji will be covered in Vocabulary anyway,
So you may learn the Kanji 食 with its kun-reading 食べる, but when you learn Vocabulary you will learn for sure the word 食べる anyway.
I personally learn only kun-readings when learning kanji if the kun-reading is not going to be learned in my vocabulary learning anyway.
You can do the same for on-reading and then not much is left anyway because nearly every on-reading is used in a word
I would say knowing the Readings of a Kanji is important, but its more important to do proper vocabulary learning as you will learn the readings of the Kanji too.
Best example for me was 飛行機, i can only remember the on-readings of those Kanji because i know the word 「ひこうき」 and it was one of the first words i learned.
When i was learning the On-Readings of those Kanji seperatly it was hard for me to remember them, my brain just dont allow me to remember every on-reading of every Kanji (even if i try hard), but thanks to the vocabulary that uses the Kanji i can remember them.
To actually know if the used reading is on- or kun-reading is more important!
So if you see a word for example 飛行機 (ひこうき) it is important to know if the readings used in this word are on- or kun-reading.
And Kun-Readings are "Vocabulary" anyway so i dont bother learning them when learning Kanji.
So my Answer to your question:
Dont bother learning all readings perfectly. If you learn Vocabulary afterwards you can drop the kun-reading completely as they are covered there.
You should try to learn as much on-readings as possible but i tell you something.
If you see a word you never seen before, its even for japanese nearly impossible to guess which reading to use as it depends on so incredible many things.
If you know the Word (written with Kanji), you know how to read it. If you dont know the word, you can just "guess" the readings and meanings, but that doesnt help you really. In the end you have to lookup unkown words in dictionary and you will see the reading then.
For me personally when learning Kanji the "meaning" of the Kanji and the way to write it (writing is veeery important to remember the Kanji correctly), i try to drop as many readings as possible as they are covered in vocabulary anyway.
If you want to become an Kanji Master knowing all readings, i recommend you to wait for that until you learned Japanese, if you are able to read/speak Japanese you know already most of kun-readings and on-readings so you only have to learn a very little then.