In "All Our Yesterdays" Spock says something like, "Arrrghhh, I've eaten animal flesh," with much regret. It indicated that traveling that far back in time messed up his Vulcan mindset. So I assume he's normally either vegetarian or all-out vegan. McCoy flashes smuggled Klingon ale at Kirk from time to time. There's one where the evil side of Kirk drinks some kind of liquor. There's one where they feed a bunch of kids ice cream. But does anyone have any more info about the crews diets? Does Kirk ever eat a pizza, a steak, a fish? Is there any other indication about Vulcan diets?
Solid food was not shown nearly as frequently on the original Star Trek series as drinks. I remembered there was a dinner party scene in "Space Seed" with Khan hosted by the Enterprise's senior officers. However, I checked, and the cast are only shown drinking, not eating in that episode.
What they are drinking might or might not be Romulan ale.
A previous question (What is being eaten by Kirk and company in this scene?) asked what Kirk and McCoy were eating in this picture, from "By Any Other Name."
The novelization did not describe these comestibles as anything more specific than "food." The unappetizing nature of these "food" items inspired this meme:
However, it was evidently considered high-quality food, good enough to serve at a diplomatic reception in "Journey to Babel."
In addition to various drinks and the strange colored balls, you can see bowls of what look like translucent rock crystals. However, toward the left there is also a tray, bearing what are clearly some ordinary Earth fruit. So there was some of what we would consider regular food to be had.
Finally, there was another dinner party—this time with Klingons—onboard the Enterprise in Star Trek VI: The Undiscovered Country, and it sure looks like Kirk was eating a mix or Earthly (and possibly alien) vegetables. There's definitely a tomato in there.
You ask specifically about Kirk and Spock, but I believe for this question we can consider food habits from other shows as well as they are all the same universe and some are not that far on in-universe timeline from each other.
Vulcans are vegeterians, possibly vegans
It is a part of their anti-violence philosophy. This is stated more than once by T'Pol in "Enterprise" series. In one episode (sorry, can't find the name) when ordering junk food on Earth she asks:
Does the Fiesta salad contain animal products?
Which means she is probably vegan.
Or T'Pol again:
You humans claim to be enlightened, yet you still consume the flesh of animals... You remain impulsive carnivores
Also Vulcan food is rather tasteless from a human point of view, which is probably also refers to their culture of being more productive, less emotional about what you do (consume).
We usually see that non-human members of the crew have possibilities to follow their cultural preferences in food (T'Pol keeps being vegan, Seven eats simple, B'elanna gets ritual Klingon dishes when needed) and we are already introduced to Plomeek soup as one of the favorite dishes of Spock in TOS. So probably Spock eats whatever correspond with his tastes and food habits.
Here is some Plomeek soup to keep Spock happy:)
Human crew members prefer same human food and drinks as we do
Apart from specific occasions when it is not available, crew members on Enterprise eat normal food - meat, fish, pies, vegetables, cakes etc. Mostly it is produced by food replicators from some kind of replicator gel. This makes food taste reasonable, but not great. Still, it's nutrition is sufficient to serve as a constant food supply.
The point here is that their food habits are the same as we are used to. Even if made from a gel supplement, people still consider that food should look and taste like "normal" food.
This is shown in details in "Voyager". Captain Janeway is coffee addict, Seven of Nine prefers tea, though she is repeatedly advised by other characters to replicate some variety of food to enjoy the taste (as apparently is what others do), Kes organizes the whole process of groving their own vegetables to cook (for real food, as being said, tastes better then replicated one). And it's the whole big story with Neelix being a cook to provide the crew with variety of home-made dishes. So again, we see that food habits of the team are pretty normal - noone expects them to eat a nutrition paste from tubes or unidentifiable food supplements.
Mind, that "Voyager" takes place not much later after TOS, so presumably Star fleet members used to normal food throughout the whole Star Trek timeline.
They drink alcohol
There are a lot of occasions in all the shows when ship crew drink alcohol, I will not even try to refer them all. It's apparently not restricted (unless you're on duty, for sure).
So what about those strange colored cubes in TOS?
In-Universe: we know that they have replicators (called synthesizers in TOS) that can produce more than just colored cubes. We see Kirk orders chicken sandwich and coffee in episode Trouble with Tribbles. So they actually eat normal food apart from those cubes.
One could think the replicators aka synthesizers are yet not developed enough to provide all those variety of food we see in later shows, but it can not be the case. We see they already use replicators on Discovery (in Star Trek: Discovery) and there are no funny food supplements. And Discovery takes place 10 years before TOS, so the technology should have been properly developed by then.
Out-of-universe: in TOS they probably tried to create some futuristic vibe using this simplified food bites. Also it is widely known they were low on budget on set.
So in general on a Star Fleet ship Vulcan eat vulcan food, human eat human food. Even in TOS it is indicated that there are possibilities on Enterprise to provide those for Kirk and Spock. The question about the colored cubes remains open (maybe they just taste good?), but I'm afraid that out-of-universe explanation is appear to be correct here.