What are the traditional Jewish opinions about the importance of learning and speaking Hebrew with proper grammar (dikduk)?
It is good and necessary for a knowledgeable person (ben da'as) to learn a little of the wisdom of grammar and to know the general rules, whether plural and singular pronouns, feminine and masculine, past, present and future tenses, direct and indirect objects, or prefix and suffix letters. But it is unnecessary to waste one's time on the various conjugations (binyanim) that grammarians discuss. Also, the points of emphasis and softness (dagesh and rafah) and the general rules are incumbent on a person [to know] as he is a man and a "nefesh chayah" as Targum translates (Bereishit 2:7). How can one who has studied, learned, and taught until he reached the level of teaching and judging (yoreh veyadin) not be embarrassed is someone asks him about the prayerbook why the vowels under "vav" prefixes change between "UNeshabeichacha", "UNefa'arecha" with a melafoos and "VEnamlich", "VEnazkir" with a sheva...? However, it is not worth wasting time on knowing all of the details and details of the details and the derivatives of the general rules because knowledge of them is a great confusion and a minor relevance. Some have explained thusly what Rabbi Eliezer said (Berachos 28b) "Keep your sons away from 'higayon'".
On the other hand, the Maharal (Nesivos Olam, Nesiv HaAvodah chapter 12) writes (translation by me):
Concerning the grammar of all the paytanim (poem-writers), the Ibn Ezra of blessed memory picks a fight, because they weren't accurate with the grammar of the [Hebrew] language in their piyutim, as he spoke at length in his explanation to Koheles (ch. 5). But I say that our rabbis of blessed memory didn't pay attention to the grammar of the [Hebrew] language like this, because the scriptures are as they were written in the holy language, and certainly this language — the holy language — can't leave the precise grammar even for a few words, and if there is anything abnormal in its language, it is because of a certain expounding. And the same is with prayer, about which they of blessed memory said (Shabbos 12b) "One who asks his requests in Aramaic, the angels don't listen to him because the angels don't know Aramaic." Certainly, with [prayers] they [also] had to use precise grammar. But with piyutim, which are like song and praise which a person says, and is a person's own speech, they didn't prevent him from using a language like this, although it bends the rules of the grammar of the holy language, since the matter can anyway be understood, because all languages are okay for this.
If so, all the more so you don't need to care about grammar in ordinary conversation.
(I found this Maharal after looking at his discussion of the piyut "Machnisei Rachamim" which we say in slichos, since it immediately follows it; it is worth looking at this time of year.)