From blue and red through heart gold and soul silver there have been many pokemon and many battle strategies as well as a few esoteric techniques. How do you play?

The Annoyer

The annoyer was introduced with Koga in blue and red versions. More or less wearing down opponents with added damage is the key to this strategy.
Poisoning with an added effect or even poison gas or toxic, evasion with sand attack, double team, minimize, protect etc.
Sometimes combos with raising stats to improve a pokemon and using baton pass to pass the stats to a normally weaker pokemon (for instance using swords dance three times with Farfetch'd and baton passing to Chansey who normally has weak attack power)
Effectively using two-turned moves such as dive, dig and fly add to the nearly untouchable annoyers who confuse, induce sleep, paralyze, curse, leech seed and poison.
Some annoyers even manage to keep healing themselves with moves like heal bell, recover, rest (usually topped with the use of a berry to wake them as soon as HP is recovered) dream eater, leech life and soft boiled, sometimes avoiding problems altogether with safe guard (even self-induced confusion with moves like outrage and petal dance).
Another touch to the annoyer strategy is to use reflect and or light screen to keep a normally easy to knock out pokemon in the game even for a little longer.
These strategies are easily beaten with mind-reader, sweet scent, foresight, the ability keen eye, the ability immunity (I think that's what it's called), shed skin ability, natural cure ability, refresh, safe guard, heal bell, quick good type match ups, brick break for light screen and reflect, and feint for protect and detect.

Control Freak

The control freak uses environments to strengthen their team and weaken their opponents. The most common way to do this is using weather moves, ie rain dance, sunny day, sandstorm and hail.
Combos with rain dance include using a normally weaker water move while it is powered up such as waterfall or bubble beam, or using thunder because it won't miss in rain.
Sunny day combos use fire moves like flame wheel, grass moves like solar beam (that doesn't take two turns in the sunlight), or even the healing moves like synthesis, moonlight and morning sun (all of which heal more in the sun).
Hail and Sandstorm both inflict damage every turn on pokemon who aren't the right type to avoid the damage. Sandshrew's sand veil even gives added evasion in sandstorms.
The problem with this strategy is no diversity with pokemon type which makes weaknesses easier to zone in on. The solution? Switch-hitter pokemon.
Switch-hitter pokemon are what I call pokemon that can adapt to weather changes or that can learn moves unnatural for their type. For instance, Houndoom is a fire and dark type. Give it a move pool that includes both a fire move and solar beam. This will allow an advantage in the sun over ground, rock and water opponents. Piloswine is a ground and ice type. It won't take damage in hail or sandstorm so it works well with damaging weather patterns. Starmie can learn thunder, solar beam and a variety of water moves. This makes it useful in both rain and shine. Castform is naturally the best friend to this strategy.
The deadliest ploy with this strategy is using moves with high power like fire blast, scared fire, blast burn, hydro pump, water canon etc to do super high damage.

Power Player

Power players focus on doing as much damage as possible and care little for strategy. Power playing like all strategies has strong points and weak ones.
Power playing can be useful with support moves. Even using one swords dance before using a powerful move like slash can greatly increase damage.
Curse is another move to be used in support (if it is not used by a ghost type). Using curse once and following through with any physical attack will increase damage.
Other support moves include leer and other moves that affect the opposing pokemon instead.
Using helping hand in a double battle on a pokemon like Machamp with a high physical attack can make your pokemon all the more deadly.
Using a pokemon with high defense and or high special defense can poke holes in this strategy

Speed Kill

There are always the lovers of one-hit-knock out moves. One hit and that's all it takes, next contestant. Guillotine, fissure, horn drill and sheer cold...
A flaw here is the hit rating of one-hit-knock out moves. 30% chance last time I checked. But what helps accuracy? Lock on and mind reader!
Using a pokemon with lock on like Magneton, switching to the one hit knock out pokemon and using the one hit knock out will guarantee a hit on the pokemon, unless the pokemon is not affected by the move. Type match ups still count (for instance a ghost pokemon will not be hit by horn drill and a flying pokemon will not be hit by fissure).
The problem with this is the damage your pokemon will probably take with switching in and out constantly.

The Balance

Of course there are the players that balance their teams. Players that teach their pokemon many different moves to give them a good type match up against most pokemon, players that use pokemon with high stats or well-rounded stats.
This strategy is the hardest to poke holes in. With so many pokemon able to learn moves they hate and normal pokemon able to learn virtually every move type there aren't many predictions for these teams.
Look for pokemon with remarkably high stats. For instance, all of Eevee's evolved forms have one very high stat, starter pokemon have one very high stat and legendary pokemon have one very high stat. If you know what that stat is (most likely an electric pokemon has high speed but low defense and rock pokemon have high defense but low special defense) work around it.
If you don't know their strategy, lead out with an electric pokemon or a normal pokemon since they have the least amount of weaknesses in the game. Thus it's unlikely that their weaknesses will be exploited.
Think logically. If they have a Charizard, then most likely Charizard knows at least one fire move. If they have a Steelix, most likely it has a ground move and a steel move.
Look for pokemon that are immune to another type. For instance Pidgeot, a normal and flying pokemon is immune to ghost moves and ground moves. Gengar, a ghost and poison type is immune to normal and fighting moves, and if levitate is its ability, it is also immune to ground moves. Steelix, a steel and ground type is immune to poison moves and electric moves.

Single Type Battlers

These are the easiest to combat in my opinion. With gym leaders, they started giving them dual type pokemon to make them more challenging and harder weaknesses to zone in on.
Know type match ups. All types have weaknesses, some pokemon even have weaknesses to the same type with both of their types.
Remember a few pokemon with diverse move pools. Pokemon like Typhlosion. Typhlosion is a fire type. But it can learn earthquake to deal with pesky fire, electric, poison, steel and rock types. It can learn thunder punch to combat water and flying types. Most fire moves are also in its pool to take care of grass, bug and ice pokemon.
Pokemon with such different types...still have weak points. Zapdos, an electric and flying pokemon has two very different types. Normally flying is weak against electric moves, but the electric type give Zapdos an edge. Electric moves do normal damage to Zapdos. Electric pokemon are normally weak to ground moves, but flying type gives Zapdos immunity. Remember that rock and ice moves still do twice as much damage against flying pokemon :)
Never forget the two types that weaken themselves. Ghost and dragon. Remembering that could prove quite useful.

Set Damage Battlers

Special Attack, Defense, Attack, Special Defense, Speed and Hit points. Pokemon stats. And always remember that each move has not only a type, but another category of physical or special. These battlers ignore that.
Dragon rage always does 40 damage.
Sonic boom always does 20 damage.
Seismic toss and Nightshade do damage equal to the user's level. So at level 40, the damage will be 40, at 100, the damage will be 100. Which, doing the math should prove that most pokemon can't withstand many of those.
There are no critical hits or damage multipliers for these moves.
These moves are perfect for getting around using pokemon with low attack or special attack and opposing pokemon with high defense and special defense. That means that reflect and light screen don't stop these moves either.
Most set damage battlers cause a status problem like confusion or paralysis to make their job easier.
Type match ups still apply so ghost pokemon will not be thrown by seismic toss and normal pokemon will not be affected by nightshade.

The Legendary Battlers

The dreaded loser that couldn't put together a fun, creative team with over 400 pokemon to choose from. They instead pick the pokemon with the highest stats in the game as well as the strongest move pools, some of them with attacks that only they can learn (I know Smeagle can learn anything it survives, transform makes Mew and Ditto able to use the move as well and mimic, metronome and mirror move bite that rule on the ass too). Here's what we'll look at.
Most legendary pokemon off the top of my head are psychic, dragon and or flying types. I'll name a few off the top of my head. Lugia, Articuno, Entei, Latias, Latios, Groudon, Mew, Celebi, Uxie, Mesprite, Azelf, Palkia, Dialga, Giratina, Jirachi.
***Lugia, Articuno. Latias, Latios, Mew, Celebi, Uxie, Mesprite, Azelf, Palkia, Dialga, Giritina and Jirachi are all either psychic, flying and or dragon types.
So on from that, play out those weaknesses! Dark types and bug types can take down psychic types easily.
Flying types are weak against ice, electric and rock moves.
Dragon types are weak against dragon and ice types. (Though Kingdra and Palkia are two dragons that aren't weak to ice because of their water type as well.)
Don't forget that a lot of them are dual typed. Lugia is a psychic and flying pokemon that can learn water moves. So a rock type might not be the wisest choice.
Remember that pokemon don't have to be of that type to learn the move. You can have a Lapras (water and ice) that knows psychic, solar beam and thunder! Gyarados (water and flying) can learn fire blast, dragon attacks and electric moves. Dragons like Dragonite can learn electric, water, fire and ice moves.

Any other battle styles I've forgotten, just add them to the bottom of the page's comments or private message me :)