One of the first things you might notice if you look up German verbs in English Wiktionary is that they include a class number right in the main label. For example the entry for sehen says "(class 5 strong, ... )". English Wikipedia explains this terminology in Germanic strong verb and it's covered in a bit less detail in German Wikipedia under Starkes Verb. I understand it's important to distinguish between strong and weak verbs because they have different conjugation endings in the past tense, for example the simple past tense of a weak verb usually adds a ''t'' (''drehen'' -> ''ich drehte'') but not with a strong verb (''sehen'' -> ''ich sah''). But I've never considered it necessary to learn the class numbers of individual verbs, despite the fact that this information is emphasized in Wiktionary.
My reasoning is that I'm primarily interested in learning Modern German, not the German that was spoken centuries ago, so unless the information is helpful with the modern language there isn't much point to memorizing it, even if it might be interesting from a historical point of view. The class number does not determine vowel changes in Modern German, for example ''meiden'' and ''greifen'' are both class 1, but conjugated ''ich mied'' and ''ich griff''. (I'm sure there was some historical reason for this difference, which may be interesting, but unless it's part of a pattern that helps to memorize vowel changes it's not very helpful.) Some other classes are more consistent, for example class 2 seem to regularly change ''ie'' to ''o''. But it seems simpler to call these "''ie'' to ''o'' verbs" that to say they are class 2 and then say class 2 verbs change ''ie'' to ''o''. Meanwhile, the vowel change does not determine the class, for example ''bergen'' is class 3 and ''brechen'' is class 4, but their vowel changes are the same: ''ich barg'', ''geborgen'' and ''ich brach'', ''gebrochen''. (There may be some subtleties with long vs. short vowels going on here, but if so, it doesn't seem helpful.) I've noticed that German language dictionaries don't include the class number; German Wiktionary gives a very reduced conjugation table compared to English, and DWDS is happy to just give 3rd person singular present and past and the past participle. (With most verbs this is enough information to fill in the tables if you know the general pattern.)
So my questions are: Should I, as a learner, pay any attention at all to these class numbers? Do native speakers, outside a few Philolog(inn)en, pay attention to them?