It’s been implied in certain Pokedex entries that only a select few Pokemon are actually capable of understanding human speech. If this is true, how is it that all Pokemon are able to understand the commands for each their attacks right after being captured, with no training whatsoever?

Difficult to explain, but I suspect they understand the intent behind words, as expressed through a combination of tone and body language, to a degree that is unusual for humans.  They may not know, right off the bat, exactly what words mean, but they can tell instinctively whether the tone of a command is aggressive, or cautious, or desperate, and they can tell the difference between being told to use a regular technique and being told to do the most powerful, dramatic attack in their arsenal.  The first couple of battles will often be rocky; that’s part of what characters in the anime mean when they talk about having to learn to work in unison with your Pokémon, and it’s part of why empathy is so often stressed as a vital quality for a Pokémon trainer.  You don’t see this in the games, of course, but, well, would you like to?  It would just be a pain.