I'm looking for a term that describes changing a word's part of speech while maintaining the root word. For example changing the verb "remember" into the noun "remembrance" or the adjective "quick" into the adverb "quickly". "Class change" comes to mind but doesn't feel correct.
Two concepts are to be clarified. These are derivation and inflection.
Derivation can change the part of speech of a word. Inflection does not.It rather maintains the part of speech of the word.
E.g.: manage is a verb. But if you add the suffix like "er", you obtain manager , which is a noun. You simply added a suffix to the stem and change its part of speech. This is derivation.
You're talking about changing the word by adding morphemes to the root. This is called Affixation, because it adds affixes. An affix is added either at the beginning of a word (where it's called a prefix, like un- in unwelcome), or at the end of a word (where it's called a suffix, like -ance in remembrance). Affix is the cover term for both prefixes and suffixes. But it doesn't cover the results of the affixation, just the fact that there was a change in the word.
Sometimes affixing something to a root changes the part of speech. But parts of speech aren't assigned to roots in English; most English roots can be used as verb, or noun, or adjective, without adding any affixes; and there are plenty of affixes that don't determine part of speech.
So there's no special term in linguistics for changing part of speech, particularly since "Part of Speech" is a very unclear term -- some say there are only eight parts of speech (though they vary on which ones they are), but others say there are many more, depending on the language, and the linguist doing the counting.
Changing a words PoS without changing anything else is conversion. This may include changing position in the sentence.
Using a borrowed word that was derived in the donor language may still count as derivation in the receiving language by stretch of the imagination, also if it was conjugated according to the donor language system, if derivation is a catch-all terminus.
conjugation is morphological construction according to the morphological rules and word-classes of a language to show modus, valence, and all sorts of exotic patterns that are not strictly covered by inflection. This may include changing Part of speech or suffixiation. See also emotive conjugation for the extend to which words correlate.
inflection as a kind of conjugation of a paradigm does not involve changing the PoS. Inflectional endings (eg. plural-s) and substitution (to be, I am, I was) are the most likely mechanisms involved.
Surely there are handbooks (Routledge, HSK, Oxford, etc.) to cover this better than I have.
Etymologically, remembrance looks like it has all of these. I'm not sure of the Old French situation before Middle English borrowed it via Anglo-Norman French. As far as I can tell, it must stem from a participle like construction, ca. in remembrance, from the OF verb remembrer. Wiktionary helpfully affords a conjugation table with the gerund en remembrant (en.WT: remembrer, -ance; PS: see The gerund and the present participle from late Latin to medieval French). This must have become reinterpreted in English, rather than calqued, because the gerund would correspond to -ing forms. Then it was like maybe understood as plural with imitative nasalized vowel and plural -s, which is not reflected by the spelling, or perhaps rebracketed to read -embrace. Also in parallel or subsequently, conversion from gerund to gerundive and finally a nouny noun seals its fate. Something like that.
PS: Modern English is generally leaning to the analytic side of the Jasperson Cycle making the fusional aspect (inflectional morohology) less relevant. Hence, remembrance does not fall under conjugation in modern English, although it does in Old French.