Are both expressions correct and idiomatic or just the first one is? Do they mean the same thing?
You should make a plan / programme for your day.
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Both are grammatically correct, but they don't really mean the same thing. It depends on context.
"Plan" is very, very general; it's your go-to for most situations, especially those on the casual side. If I'm writing down which chores I'll do today, I'm "making a plan for my day"; if my friend and I are deciding to go see a concert together, you would say we're "making plans" to go there; et cetera. It would sound extremely weird to use "programme" for those situations.
A "programme" is much more formal and organized; it's for things released by large organizations, particularly entertainment, performance, and tourism. You might refer to a specific show on TV as a "TV program"; and if you go to a play or choir performance, the little booklet they give you that lists the names of songs and when intermission will happen is called a programme. You would absolutely not say "TV plan", though!
There are some situations where either word may fit, though it'll still have particular connotations; "military plans" and "military programmes" both sound right, but "military plans" would sound more like planning out tactics, while "military programmes" sounds more like rules & regulations, bureaucratic sorts of things.
Finally, there may be regional differences at play as well. "Programme" is the British spelling; here in the USA we spell it "program", and tend to use it less overall (i.e. we just say "TV show", rarely "TV program").
This isn't a comprehensive list of all the differences, but hopefully a good enough overview of the basics. Hope it helps you!
Both expressions are correct. Neither are idiomatic.
An idiom is a phrase or expression that typically presents a figurative, non-literal meaning attached to the phrase; but some phrases become figurative idioms while retaining the literal meaning of the phrase. Categorized as formulaic language, an idiom's figurative meaning is different from the literal meaning.
These phrases are literal, at least in the zero-context situation provided in the question. They are asking you construct a plan or programme, these words are synonyms for writing out a list, of things you would like to do with your day.
An idiom requires some non-obvious context to make sense.