When should I use archaic and obsolete words?
Solution 1:
When should I use them, should I use them at all?
Probably never, unless you're writing historical fiction. Archaic and obsolete words are words that are no longer used in contemporary society, so unless you want to specifically emulate olden times, it's best just to leave them alone.
What's the difference between these descriptions?
According to the Standard English section of the M-W preface, archaic words are older, perhaps at least a century out-of-date and used only for a deliberately old-fashioned effect in modern times, while dated words went out of style more recently. Historical words are words that are still used, but only to refer to ancient things. Rare words are words that are slowly leaving the English language.
I don't see obsolete as a usage marker in my edition of M-W, but I would roughly equate it to being between archaic and dated. Obviously, your dictionary's preface should explain how obsolete would be used by their editors.
Also, should I spend time to remember these archaic and obsolete meanings?
Probably not, unless you're reading a lot of historical stuff. You can always look them up in a dictionary if you're only going to encounter them once in a while.
Solution 2:
You should use obsolete or archaic words when:
No other word will serve (as in a scholarly piece about history or linguistics, for example).
You want to confuse your audience or make them laugh.
You want to sound pretentious or pedantic.
Solution 3:
I wouldn't say that no one should ever use archaic words; sometimes you do need a particular nuance. But I would say that somebody learning a language (any language) is not well-equipped to judge these cases, so it is better to stick to the simpler, more-straightforward word choices until you are more comfortable with the subtleties of the language. Every time I have tried to use an "advanced" word in the second language I've been learning, I've made things worse rather than better.
Solution 4:
Archaic means that a word has the flavor of old-timey language, and brings the feel of the past along with it. Archaic language is generally used infrequently and for effect when it is. I think that in English-speaking countries, the most common uses of archaic language are when religious texts are being quoted, or when older literature is being quoted--or performed, as in the case of Shakespeare or other period plays. Here's a fuller definition at Merriam-Webster.
Obsolete is a cousin of sorts to the word "archaic", and using an obsolete word will certainly make what you're saying seem esoteric and strange--but an obsolete word is a word that's no longer used at all. One should use such words with extreme caution: people listening to you or reading your writing may not know what you're talking about, unless you very carefully provide good context for them to figure out the meaning of what you're communicating.
I wouldn't prioritize learning obsolete and archaic words until you get more advanced in English, and I don't think anyone's going to fault you for failing to know archaic words.
If you want to take a crash course in archaic language, dipping into the plays of William Shakespeare would be a good start, as they are readily available, but prepare to be extremely bewildered! There exist translations of both of these with notes to explain the more perplexing turns of phrase.