Please explain why this sentence is grammatically incorrect.
I'm going to take and stir the cake mix.
There is nothing wrong with it grammatically. "Take" is just a verb applied to the cake mix, and you can apply lots of verbs in one sentence.
However it does sound odd, and that's because there is a more commonly used sentence structure that's used with take. Take is such a common word, that it has these kinds of ultra-common uses. When we're accustomed to hearing a word used in a particular way, it sounds odd to hear it phrased a different way. It makes us think "Huh? Why didn't they phrase it the normal way?"
The common structure is:
Note that we have and twice in these sentences, because it fits that "take it and [verb] it" structure, but you could drop one and if you like:
Also note that most of the time the take is a redundant part of the phrase in any case. You can't stir the cake mix without taking hold of it, so that part is implied even if you don't say it. There is really no need to use the word take here, except insofar as it becomes a familiar and natural sounding sentence.
The take and + Vb construction in English is one of a number that fall under the general heading of Serial Verbs (known also as "small verbs", in the case of English). Take is a causative of go, a verb which occurs frequently in such constructions.
Serial verbs occur when two verbs whose meanings complement each other are used together in a single verb phrase. Some languages frequently form complex serial verb constructions, like go cut carry stack wood.
Other English serial verb constructions include
See question 4 on this Midterm Exam.
Grammatically, there's nothing wrong, but there is something slightly odd about this usage of "take". In "recipe" contexts we often see "Take three apples, two pears, etc., [and cook like this]". We do a bit of a double-take in OP's sentence because we're expecting one or more nouns (ingredients) to follow "take".
The word "take" has very little meaning in such contexts, because the cake mix isn't actually being taken to anywhere, or taken from any particular place or person. We wouldn't just say "Take the cake mix" as a sentence on its own here. It's already present - we're just calling attention to it. Thanks to @ruakh for naming the role of "take" here as simply to "topicalise" "the cake mix".
In this construction, take (or get) is really just a "placeholder" verb introducing the subject noun (cake mix) - so long as the noun follows immediately we don't need to consider what if anything "take" means. If instead we meet "and" and another verb, we're encouraged to treat "take" as a meaningful independent verb - but it isn't, which is why OP's sentence sounds "awkward".
Taking similar examples common enough to be meaningfully contrasted, "take and compare this" gets only 2 hits in Google Books, where the more "natural" "take this and compare" gets 191. And "bring and show it" gets 9 hits, compared to 791 for "bring it and show it".
To take a more "unexceptional" conjoining of two meaningful but relatively disparate verbs that aren't so awkward, consider the 236,000 written instances of "buy and read" in Google Books. Even more relevant, 2470 instances of "take and read this", with "take" used as a meaningful verb (i.e. - physically take this text away with you, and read it later).
John Lawler refers to this as serial verb usage, but I'm not convinced English really has such a construction. And if it does, it seems more like e.g. "Look what you've gone [or "been"] and done!".
From the comments/votes here it's clear not everyone sees anything unusual in conjoining two disparate transitive verbs with "and" before naming the noun to which both refer. Personally I find it clumsy unless the context and/or inherent meaning of the verbs make it natural to conflate them into a single integrated action.
In OP's example, "take" does little more than indicate that the speaker is about to introduce the noun subject - but instead we get another verb, which grates on my ear.
Here's a completely different angle on this thread: take and is a quirky little colloquialism in Wisconsin. As such, it misses the point to discuss whether take functions as a legitimate transitive verb in expressions with take and. Once you’ve heard a person who routinely inserts take and into sentences, it puts the take and stir example in that specific light. Here are a few examples I've heard:
In plain English, in this context, it (i.e. the verb take) takes an object.
That is to say, the verb "take" takes, i.e. is required to be followed by, an object. It is functioning as a transitive verb.
So, a correct rendition would be: "I'm going to take the cake mix (object) and stir it (i.e. the object - cake mix)."
Note: "take" can also be used in an intransitive form.
For example: John takes but never gives. (No direct object - taken, given, or needed.)
The sentence, "I'm going to take and stir the cake mix", sounds weak to my ears. Though I'd advise against that construct, I'm not sure if I would call it ungrammatical.
The reason it sounds odd has to do with Thematic roles (called Semantic Roles here). In the sentence discussed, the same phrase, the cake mix takes on the role of the Theme of take and the role of the Patient of stir. However, in the sentence "I'm going to take the cake mix and stir it", it is the Patient of stir, and there is no shared object.
Here are a couple of examples that still use conjunctions of verbs, but where the shared object does not perform a different thematic role for each verb:
These don't sound odd to me. (Assuming I had good reason to gift cake mix. That I would attempt to beat cake mix is probably an indication of how much I know about cooking.)
=== Exercise left to the reader ===
VerbNet has a fairly large list of verbs, where they have listed out the thematic roles associated with each verb. Pick combinations of verbs and check if the semantic roles of their objects match. See if this agrees with your own intuition of which verbs can be combined with one of and, or and but.
This sentence is not grammatically incorrect:
The construction is just not done in the English language.
You will find examples of such construction, in fact the exact sentence, in at least a few other languages, though.
Every grammatical construction -- even one that makes perfect sense -- need not be usable in English, for want of familiarity of usage.
You can't use "and" to glue two things together into one thing. The sentence tries to make "take and stir" act like a unified verb that gets a single object.
It's the same reason you can't say "I'm going to take the cake mix and oil". You have to say "I'm going to take the cake mix and the oil". You can't glue "cake mix and oil" together to make one object which gets one "the".
You can't glue "take and stir" into one verb that gets one object. English simply has no rule to allow that.
When combining verbs in a sentence like that, the verbs have to be in the same tense. In the sentence, "going to take" is future tense, while "stir" is simple present. I believe that in the case of "going to", it is actually allowed to omit "going to" for the second verb, but that would imply, that the two actions happen at the same time. Here, clearly, the mix will not be stirred, until the "taking" of it is done.