Some of my students have a disagreement about transcribing the pronunciation of "trade" in American English. Some say it's (a) [t͡ʃeɪd] while others (and they point to dictionaries that support them) say it's (b) [treɪd]. I thought it was the former; the latter sounds a little like a fast "tirade". But now as I try to articulate it more, it seems like (c) [t͡ʃreɪd]. Any insights on this?
The answer to your question depends to a large extent on whether you are using IPA to represent a phonemic transcription of the word, or a phonetic transcription.
In either case, your option (a) is absolutely not correct. IPA /t͡ʃeɪd/ (or /t͡ʃɛɪd/, depending on how you choose to transcribe diphthongs) would as regularly as possibly be represented orthographically as ‹chade›, even though this is not a word, as far as I know. A minimal pair can easily be found, though: ‹chuck› /t͡ʃʌk/ vs. ‹truck› /trʌk/ (phonemically).
If you wish to transcribe the word phonemically (between /slashes/), according to the underlying phonemes it contains, John Lawler’s comment is immensely pertinent: there is no phonemic opposition in English between initial /tr/ and /t͡ʃr/. Whether the sound cluster found initially in ‹trade› is underlyingly made up of /t/ + /r/ or of /t͡ʃ/ + /r/ cannot really be determined.
As such, it doesn’t really matter which one you choose to use, as long as you do so consistently. You can write /treid/ or /tʃreid/ as you see fit. The former is probably more likely (adhering to Occam’s razor) and is what is normally used in dictionaries and other works that include phonemic transcription; but the latter is at least a theoretical possibility.
If you are rather intending to use IPA to transcribe the word phonetically (in [square brackets]), describing in detail the actual sounds that your speech organs produce when saying the word, it is irrelevant what underlying phoneme the word is made up of—but it becomes then necessary to define which pronunciation of the word you are talking about. Some dialects will have differing pronunciations of more or less any word in any language. In this particular word, though, the main differences are twofold:
The quality of the phoneme /r/. Some dialects (most forms of BrE, SAE, NZE, and variably AusE) have a simple, alveolar or retroflex approximant [ɹ] or [ɻ]; others (especially most forms of AmE and IrE) consistently labialise this and have [ɹʷ] or [ɻʷ]. The exact place of articulation varies, but it is usually retracted enough to count as at least semi-retroflex. (Some, like Scottish, tend to have an actual [ɾ], but let’s leave those out of this altogether.)
The quality of the first part of the diphthong /ei/. Some dialects have [eɪ], others have [ɛɪ], some even have [æɪ] or [aɪ] (most notably AusE), and some (like Scottish and Irish, and some BrE dialects) monophthongise the sequence to [ɛː] or [eː].
If we focus on AmE, the first is almost always [ɻʷ]; and the second varies, even from speaker to speaker, between [eɪ] and [ɛɪ], but this difference does not make a difference for the initial consonant cluster, so I’ll just be lazy and write /eɪ/.
The transition from the plosive /t/ to the approximant /r/ causes the cluster to become an affricate, which is an almost universal case of assimilation. Since in AmE the /r/ is nearly always retroflex or near-retroflex, the assimilation causes the /t/ to move towards a retroflex place of articulation, i.e., the tip of the tongue is held against the hind part of the alveolar ridge and the tongue moves backwards upon the release of the plosive: [t͡ʂ].
In contrast, the phoneme /tʃ/ is postalveolar or palato-alveolar, meaning that the blade of the tongue (the part just behind the tip) is against the front part of the alveolar ridge, while the dorsum (the back of the tongue) is domed upwards towards the hard palate. The tongue moves forward upon release.
In other words, an actual phonetic sequence [t͡ʃɻʷ] is quite difficult to produce: it involves having the tongue start at the front of the alveolar ridge, with the dorsum domed upwards, and then upon release move backwards and curl upwards, i.e., both moving the front of the tongue and switching the curvature from downward to upward.
It is much easier to pronounce [t͡ʂɻʷ], which is also what initial /tr/ in AmE quite distinctly sounds like to me.
So in purely phonetic transcription, I would write ‹trade› (in AmE) as [t͡ʂɻʷeɪd].