Suppose I am playing a game in which Agility, Dexterity, and Speed all exist (separately) as stats, and that allows some measure of control over stat growth. How should I decide which stat to invest in?

Also, is there any game that has at least two of the stats Constitution, Endurance, and Vitality?

maybe read what they do?

Your play style should determine this, and as Orkhepaj says, read the manual, look at what they do and decide which one you want.

As for your second question, yes, there are games that have two or more of those in (or synonyms of them). I believe that the Dark Souls games as a start have Endurance and Vitality in them.

What if the manual is vague or non-existent? What if the manual isn't trustworthy (for example, if I've verified that some of the manual's claims are actually incorrect)?

In most games that use them, speed is the more potent of the three with the common designs. More actions -> More power better than the others.

Grim Dawn: Constitution is your "health regen" meter that decreases as your health auto-refills when not in combat. It is filled by visits to town or random drops. It's almost never an issue except in a couple zones with significant environmental damage. Vitality is a damage type [comparable to "necromantic"] in that game.

D&D 3e Unearth Arcana 'Vitality and Wound Points' variant: Constitution is still your basic stat. Vitality is part of the alternative HP system. https://www.d20srd.org/srd/variant/adventuring/vitalityAndWoundPoints.htm

Then experiment and roll with it; go for the one that sounds most fun.

After all, we're only talking about video games, not things that actually matter.

Increasingly games have tended to boil these down to a single stat, but it's not completely unusually to see speed as a standalone (typically governing frequency of actions or movement distance in turn-based games). Someone might be fast but wildly inaccurate.

The definition much less function of dexterity vs. agility is a bit harder to guess what it might indicate. If I had to guess agility governs dodge and dexterity attack accuracy and/or possibly ranged damage. But that's a total guess.

This question is meaningless without the context of a specific game. Every game's mechanics are different.

Vitality generally means health points.
Endurance generally means stamina points.
Constitution is a combination of those with strength as defined:
"a person's physical state with regard to vitality, health, and strength."

Multiple games offer the basic health, stamina, strength point values.

Depends on the game?

In my experience from various games:
- Dexterity = Hand/Eyes Coordination. Good for hitting things or intercepting them with your hands
- Speed = Reflexes and ability to chain actions quickly
- Agility = Balance & fancy whole body acrobatics (ex: back flips)

You wouldn't be much of an action hero without all 3, but if you gotta pick and those values are taken at their extremes and the goal of the game is to kill the other guys before they kill you, I'd say: Speed > Dexterity > Agility for melee combat and the same for ranged combat except in situations where dexterity allows you to greatly increase your shooting distance and you are battling on a fairly open terrain.

However, in most games, there are laws of diminishing returns with stats so you usually don't want to put all your points in a single one (usually, but not always).

Also, there can be mitigating circumstances. For example, the value of agility increases greatly if you are fighting on a narrow ledge on top of a cliff.

I think the value depends on the larger mechanics. In an older style game, or even a modern single-turn structured game, yes having 2 attacks where everyone else gets 1 is a big swing.

But in a more real-time or continuous-turn structure it might only translate to a 10% increase in frequency of attacks, so the tradeoff of an attack that hits 10% harder vs. 10% more actions may be less clear which wins out - whereas for a support character more actions probably shakes out as the clear winner.

As Jorev notes, it's hard to say whether this is inherently good/bad, but how it is balanced within the scope of the large game mechanics.

Edit - but yes, in the context of when Wasteland was made, initiative does make more sense than frequency of action. That said, if ranged resolves simultaneously, does speed have any value at all, or is it only useful for melee?

That's not always the case.

In Final Fantasy 10, for example, physical damage scales with Strength cubed, while magic damage scales with Magic squared. As a result, the higher the stat, the bigger the bonus from getting another point. (This is also part of the reason why offensive magic is pointless in the postgame.)

(Speed is somewhat odd in FF10, as there are certain breakpoints where one point makes a huge difference, but then you can get a significant difference without any change.)

Or, we can look at Advanced Dungeons & Dragons, which has some really strange rules regarding stats. There may be no difference between 6 and 14, yet the difference between 18 and 19 can be massive (it is with Strength, which also has those weird "exceptional strength" rules).

I could also mention something like Chrono Trigger's Magic Defense. Magic Defense is a percentage reduction; the difference between 10 and 11 likely won't be noticeable, while 99 means you take half the damage you took with only 98. In other words, increasing the stat is more useful when it's already high. We also see that with physical damage reduction in Stranger of Sword City, where it's likely not worth optimizing until you can get it close to the limit of 90%.

I believe it's only useful for melee in Wasteland.

Your characters can do lots of damage in melee, so enemies die quickly if you can get to them; the problem is that getting close enough to enemies that use ranged attacks can be a problem, and even then you still get hurt (and possibly knocked unconscious) before you get a chance to act; no amount of Speed will allow a melee attack before the enemies get their shots in.

Incidentally, in games with Wizardry or Dragon Quest style combat, there are situations where you might not want high initiative. For example, if you're fighting a boss with a strong multi-target attack, and you have a strong multi-target heal you can use each round, you're better off always acting after the enemy rather than sometimes before. (SaGa 2's late game bosses are examples of this, and in the original version, due to a bug there's no way to consistently outspeed the last couple of bosses.)

The Fromsoft games (Dark Souls) have both Endurance and Vitality.

And Agility, Dexterity and Speed all mean different things. Agility generally encompasses Dexterity, but not always. Also, speed and agility are two wildly different things. A football player (American football, for this example) can have great agility, which is the ability to avoid tacklers, change direction, and/or juke, without affecting their speed. Speed is the ability to move quickly, as in their 40 yard dash time. One can be fast without being agile, and dexterity is more a stat that would affect catching and ball control.

There are 3 things to consider.

1. What the stats actually do
As mentioned by others, different stats mean different things in different games. In RPGs, speed can determine how long it takes your ATB to fill or just who acts first in a fight while agility is usually dodge-related.

2. Your playstyle
If more action oriented, speed is probably related to movement speed while agility has to do with i-frames gained from a dodge roll. Dexterity is probably the stat that determines if you can equip a certain weapon or not.

3. Stat caps
Probably the most important thing to consider. No point dumping points in speed if the soft cap is 20 and you are already at 20.

While kind of a cool mechanic, strength was badly over-powered in the second edition for fighting types because of the percentile values (and mages were total weaklings at level 1, but started being completely overpowered the moment they got their hands on 3rd spells and above... unless the DM was really stingy with spell access in the campaign... by the time they got 5th level spells, it wasn't even a semblance of a contest anymore if the person playing the mage was creative at all with his usage of spells).

Each percentile category was essentially worth another full point in another stat and races that had +1 to strength could jump straight to 19 and pwn fighting types up of 3-4 levels above them. They were really going for that Conan vibe I guess.

And yes, the 8-14 range was the dead zone.

It wasn't balanced at all, but it was kind of fun though.