According to the memory alpha article Warp Core, one part of the warp core was visible in engineering--the "dilithium crystal converter assembly", "which consisted of two flattened rounded nodules situated directly in front of the warp plasma conduits to the warp engines, which were behind a large metal grate". The Star Trek: The Next Generation Technical Manual would later establish that the matter and antimatter were actually channeled into the dilithium crystal itself, where they would react in a non-explosive way somehow made possible by the unique nature of dilithium, and the high-energy plasma that resulted from their reaction would be channeled to the warp nacelles (that's what the "warp plasma conduits to the warp engines" in the above quote is presumably talking about). The function of the dilithium crystals (referred to as 'lithium' crystals in some earlier episodes) is not really stated in TOS, although in the episode "Mudd's Women" after several crystals are overloaded, Spock says "The entire ship's power is feeding through one lithium crystal" and Scotty says "it's frustrating. Almost a million gross tons of vessel depending on a hunk of crystal the size of my fist." So, it seems plausible that the creators of the original series would have had some notion that the crystals were closely connected to the matter/antimatter power generation system.
The article on the assembly shows a shot of this part of the warp core, from the episode "Elaan of Troyius" (where Scotty actually used the term 'dilithium crystal converter assembly'):
Note that although in this scene the chamber with the dilithium crystal is exposed, most of the time it was retracted between the "two flattened rounded nodules" described above, which you can see better in this shot from "Day of the Dove":
There's a photo in the U.S.S. Enterprise Owners' Workshop Manual (which was written in consultation with Michael Okuda, who with Rick Sternbach was the technical consultant on all the shows from TNG onwards, so I think it can be considered to have almost the same level of canon as the TNG Technical Manual which was written by Okuda and Sternbach) which shows these two rounded things and calls them the "dilithium-focused matter/antimatter integrators". The text also says "Before 2267, the dilithium crystals were located in a separate control room. However, during a layover at the end of 2267 the Enterprise NCC-1701's main engineering room was redesigned and the twin matter/antimatter integrators were fitted in the centre of the room. These redesigned units contained the dilithium crystal converter assembly." Presumably the out-of-universe explanation for this is that they redesigned the engineering set somewhat, around the time of one of the later episodes that have since been defined to have taken place in 2267--this site on the history of changes to the engineering set suggests it may have been around the time of the episode "The Doomsday Machine".
As for the rest of the warp core, where the matter and antimatter would be channeled up towards the dilithium crystals, this apparently wasn't shown, but there's a note in the Warp Core article saying:
Of the original Constitution-class warp core, only the dilithium crystal assembly and the plasma conduits were ever seen in Star Trek: The Original Series. When Doug Drexler was called to design the detailed schematics of a Constitution-class starship, he designed a horizontal warp core that runs two decks below main engineering. The schematic made a prominent appearance on screen in ENT: "In a Mirror, Darkly, Part II". Star Trek: The Animated Series also featured a vertical component of the warp core, that extended from the dilithium crystal assembly.
Here is the schematic of a Constitution class ship designed by Doug Drexler (also mentioned in Bill Teel's answer), which I suppose we can consider canon since it did air on that Enterprise episode. And below is the same image with labels, from an archived copy of Doug Drexler's old "Drex Files" site (linked on the memory alpha Constitution class decks article, which has a more detailed guide to what's what)--just click the image and use the magnifying glass tool on your browser to see it full size.
And here is the image of the vertical warp core from the animated series: