What is the order of priority in Second Language Acquisition (SLA) for a beginner between listening, reading, writing, grammar, and speaking?

Over the last30 years there has been an intensive study of how a second language is learned, but as yet no-one has been able to come up with a comprehensive theory. For this reason there is also no consensus on how language should be taught.

Clearly, no-one can claim to speak a foreign language unless they have mastery of both its essential grammar and its essential vocabulary.

Some students, especially those with native languages similar to English, can quickly acquire the interpersonal language skills of speaking and listening. Research has shown however that it can take more than5 years before the non-native speaker is operating at the same level of academic language competence as his or her native English speaking peers. It can take an especially long time for those students to catch up academically whose main priority in learning English is to make friends and feel comfortable in the school. When they have sufficient English to do this, they may consciously or sub-consciously decide that they have learned all there is to learn, and "switch off."

I want to give a concise answer to your question. You can't learn a language without knowledge of its grammar ; without knowledge of grammar and sounds (phonology) , you can't understand a written or spoken discourse. Thus,  I believe that to catch a language system, you need first to DIGEST its phonological, grammatical and lexical systems before moving to comprehension of the language in question. I may even say that without ALL the "priorities" you mentioned, learning a language will be impossible.

Another approach to this question can find answers in Noam Chomsky's explanation of Language Acquistion which states that in the process of learning a language, children LISTEN to the Target language and go through some processes of memorization, generalization and differentiation before fully ACQUIRING the language.