Learn English – slang/idiom for “Get someone into trouble?”

"Get the basics straight" and "Get the basics straight first" are valid English, but are not idioms. In a context where an author is trying to:

• Be polite (perhaps sarcastic, but neither rude nor obscene nor profane)
• Point out the basics of something, before pointing out more advanced features
• Write clearly
• Use the second-person imperative

"Get the basics straight" or "Get the basics straight first" is a natural thing to say.

Here are some alternatives that are idiomatic. Notice that some of these alternatives are quite rude, or compare the reader to a baby, toddler, or young child:

• You have to crawl before you can walk.
• You need to walk before you can run.
• Learn the ABCs of <topic>
• Get your shit straight.

Here is an idiomatic alternative that is polite:

• Start with the basics.

I think what you're asking is "Is 'in layman's terms' formal enough for an academic paper?" The answer is yes, it is OK to use that expression in your paper. It is not informal or slang.

As Alan recommended in his comment 'in layperson's terms' is a good alternative if you want to make an effort to use gender-neutral terms. I personally think that layperson is a little awkward, but it can be important in some environments to choose more 'inclusive' words.

I would write "I will explain my conclusions in layman's terms." In my opinion, you should not say 'explain terms' if you are talking about certain technical words, you should say 'define terms'; I'm not sure everyone agrees with me about that. You could explain the 'terms' of a contract, but I think that is a different meaning from the one you intend to write.