What is the point of Speed-changing moves in Pokemon?

Solution 1:

In competitive battling the most common three moves that influence speed only are:

• Trick Room (Reverse initiative for 5 turns)
• Tailwind (Double your team's speed for 3-4 turns)
• Sticky Web (Any entering non-flying non-levitating mon on the opposing side will get a speed drops)

All of them have the same idea - you set up the speed-up/slow-down with one Pokemon and when it faints you summon your sweeper (a Pokemon with set of moves / ability / item whose purpose is to defeat all or almost all of enemies Pokemon).

The reason it works is that in most cases there is a big tradeoff between power and natural speed. For example, if you know your Smeargle will setup Tailwind or Sticky Web you can invest all EVs into attack and health which makes it that more likely that your Pokemon can outlive any other while still outspeeding.

A similar point can be made with Agility or Rock Polish - if you already have a Pokemon that can one-hit-ko or two hit KO an opponent, setting up being the first to hit will be useful in the long run.

For example, let's say your opponent has a Fire/Flying Pokemon out and you send out a Rock type haymaker. You can be almost certain that your opponent will switch out, meaning you have a free turn to set up a Rock Polish.

The reason it's not as popular in competitive battling is that setting up one boost to attack and one boost to speed with Dragon Dance is much better than two boost to speed only - it can transform a two-hit-ko into an one-hit-ko which takes us to the final point:

Action Economy - if you get to take a turn while your opponent doesn't, it's severely beneficial. It's what happens if you either OHKO or put your opponent to sleep when you're faster. (In the second case, in a way you both skip a turn but it's still beneficial if the opposing Pokemon doesn't wake up immediately)

Solution 2:

You are correct in that, in a 1v1 fight without switching, using Agility or another move on a slower Pokemon and then becoming faster on it does not affect the exchange of moves. As you said, the Speed advantage is negated by the turn you spent using said move. However, Pokemon battles involve two teams. Though your Pokemon used Agility to gain a Speed advantage over that Pokemon, it may now also have a Speed advantage over other Pokemon too. This can provide more benefit to using the Speed move than simply that one turn, gaining multiple turns, including key turns that can be the difference between getting a KO and being KOed. Consider the following examples:

• Jolteon vs Golem. Turn 1: Jolteon uses Double Kick, Golem uses Rock Polish. Turn 2: Golem uses Earthquake which scores a OHKO (one-hit KO). Opponent switches in Infernape. Turn 3: Golem uses Earthquake and scores a OHKO on Infernape. Due to using Rock Polish on turn 1, Golem gained a speed advantage over both Jolteon and Infernape. This gave it two extra turns (one against Jolteon and anotehr against Infernape), at the expense of one turn of using the move, effectively awarding the player one net extra turn. This is very notable in the Infernape scenario, since otherwise Infernape would move first and potentially OHKO Golem instead. This scenario occurs frequently with "sweeper" Pokemon in singles, since by outspeeding other Pokemon, it is easier to finish them off before they attack, potentially winning you the game if none of them can attack before being KOed by your faster Pokemon.

• Jolteon vs Golem. Turn 1: Jolteon switches out to Gyarados, Golem uses Rock Polish. If Golem had instead used Earthquake, that attack would have not affected Gyarados. Then next turn Gyarados could threaten Waterfall or another Water-type attack against Golem. But because Golem used Rock Polish, it can now outspeed Gyarados and threaten it first with a super-effective Rock-type move. That is one of the risks of using offensive moves, in that you may not get much benefit from doing so if it deals little or no damage to the opposing Pokemon, whereas Rock Polish Speed boost remains the same regards of the opposing Pokemon.

With moves like Cotton Spore, the opposite scenarios apply, where that the opposing Pokemon may then be slower than multiple of your Pokemon, which may cost them multiple turns. Moves like Cotton Spore are generally considered less effective than Agility though, since your opponent can remove the nerfs by switching out (whereas you can keep your Speed boosts by not switching out).

The above points were focused on single battles, which I'm assuming you are focusing on as you said you haven't played many competitive / PvP battles (official Pokemon tournaments are generally double battles). The same point does apply for Doubles, with one Pokemon now being faster/slower than multiple Pokemon. However, moves like Agility and Cotton Spore are generally less effective in doubles, since it is harder to take advantage of the boost (by virtue of there being four Pokemon on the field instead of two, which makes it harder for one Pokemon alone to establish board control). Competitive players instead tend to use moves like Tailwind and Trick Room and Tailwind instead in these settings, which affect multiple Pokemon simultaneously.

Solution 3:

A strategy that hasn't been mentioned yet is trying to make the opposing Pokémon flinch. Flinching can be a powerful effect because it completely prevents the opposing Pokémon from taking any action, and can be inflicted by several moves and two items, which means any Pokémon has potential access to the strategy.

However, a Pokémon will only flinch if it moves second, which means the strategy only works if your Pokémon is faster. Having a speed-increasing or opponent-slowing move can increase the number of Pokémon that your flinch strategy can be employed against, and is therefore worth the move slot and turn needed to use it.