With the press of a button, I need to play an intro sound file, then one of 50 other sounds in order, each time a button is pressed I need to play an intro file and one of 50 different files after it in order each time the switch is pressed, with a short delay between the intro and next file.  

For example let's say the intro is called "intro.wav," and each of the 50 other sound files are just named 01.wav-50.wav.

The logic would be:

TRIGGER
PLAY intro.wav
250ms DELAY
PLAY 01.wav
END


TRIGGER
PLAY intro.wav
250ms DELAY
PLAY 02.wav
END


TRIGGER
PLAY intro.wav
250ms DELAY
PLAY 03.wav
END


and so on, until all 50 files are played, then start over.

The messages are going to be a curated group of words. I'm not sure what the intro will be yet, but it will just be something like:

intro.wav = "The word of the day is"

01.wav = "help"

02.wav = "climb"

03.wav = "think"

I'm looking for some guidance on where to start with this, if an Arduino product would be suitable, and if so, which one.  I've worked with the Adafruit Audio FX Sound Board before, and that can almost do what I need it to do, but not by itself. It looks like I'll need an Arduino of some kind to accomplish this, so I went ahead ordered an Uno.

I'm not married to the idea of having the intro and word files separate either. If I need to create all 50 files to contain the intro and the word, that would be fine, as long as each one is triggered sequentially. 

In that scenario, 01.wav could be "The word of the day is ... help" and the button would basically be the "next track" command.  I don't really have a problem with doing that because I'm all about keeping things simple and I don't think I'd run out of space, but I think it would be less time consuming to just dump all the individual word files onto the card.

The important thing is that the track(s) would play once, then stop, and when the button is pressed again, the next one plays.
You will find all the code and thelibrary you need on the Adafruit learn center.
For such a project I would use a DF-robot (clone) MP3 player (~4$), an Uno, a speaker and an SD card. The library for the player includes examples to get you started and the sound project you plan is possible even with low programming skills. Look at the example codes provided and start to program.