One strategy that works very well in almost any Pokémon game is simply using only one Pokémon in battles. While you're supposed to be using a balanced team to counter the Pokémon that are strong against your starter, by simply using the same 'Mon over and over, it becomes so powerful that even super-effective hits deal almost no damage (and that's assuming they manage to hit you at all). Not until Pokemon Black And White was this strategy finally nerfed; your experience gains are based on the level difference between your Pokémon and the Pokémon you're fighting, so training just one Pokémon isn't really viable, as they'll be getting crap for Experience in no time. It's important to note that you WILL get a lot more EXP at the same level though, making getting early levels much easier. And later in the game, you can still be so overleveled that you don't NEED more... It's not uncommon to see a level 43 or so Emboar (usually used because it has a great movepool and decent all around offenses) before the fourth Gym, when the Gym Leader uses level 30 at the most.
The single easiest way to get through any game is by having a friend who has already beaten the game and is willing to trade powerful mons. The first Gym Leader's level 15 or so Pokémon aren't exactly a match for a level 90 juggernaut, are they? To attempt to prevent this, the designers include the badge system; Pokémon not caught by the player will usually ignore them (doing things like sitting around, even sleeping in the middle of the fight) if they are above the level designated by the player's badges. The problem is that: 1. the Pokémon only has to listen to the trainer once, while in the meantime the opponents aren't capable of doing anything at all thanks to the super-Pokémon's ridiculous defenses, and 2. the badge system seems to think that the Trainers are much stronger than they are (for example, getting the fourth badge makes a traded Level 50 Pokémon listen to you, whereas the final boss has Level ~60 Pokémon in an average game). Even easier is if you have a friend further in the game who has a bit of patience. Trade them your Mons and have them level them up on the higher-leveled Pokémon late in the game (along with the boosted EXP for trading), and then trade back when they're sufficiently leveled. Then you won't even have to worry about the Pokémon ignoring you occasionally, as your own will listen to you no matter which badges you have.
The Mew Glitch in Pokemon Red And Blue, which allowed essentially any Pokémon in the game to be caught far before they were normally encountered. Also the Cinnabar Island glitch and its associated extremely overpowered Pokémon, as well as the use of the Missingno version of that glitch to duplicate items like Rare Candies.
'Tweaking' in Pokemon Diamond And Pearl also fits, letting players catch event legendaries without the required item downloads. Game Freak fixed it in the international releases.
Also the Sketch and Mimic glitches, which allowed certain Pokémon to use the move Sketch to permanently learn almost any move in the game.
If you battle competitively, RNG abuse is probably your best friend. It allows players to use computer software to do some calculations, then set their DS's game clock to the right point and locate an exact right moment to catch or hatch flawless stat and even shiny Pokémon with relative ease. However,no external devices ever touch the game itself, meaning that many players do not consider it hacking.
The Nugget glitch in Pokémon Fire Red and Leaf Green, which allowed players to get as many nuggets as desired by intentionally losing to the final trainer on Nugget Bridge, the guy with the nugget, then battling him and losing again.
The Pokemon Emerald Cloning Glitch, an easy way to get multiple master balls, rare candies, ect. Unfortunately, it's only possible after beating the Elite Four.
Gen IV has things like the GTS trade glitch and GTS clone glitch that also can make gameplay a lot easier.
Pokemon Stadium. Go to Pika Cup with a Level 20 Charmeleon, Dratini or Arcanine that knows Dragon Rage and see what happens. It got nerfed on Stadium 2's Little Cup, though, where SonicBoom and Dragon Rage will always miss.
Wonder Trade in Pokémon X and Y can become this, if you get someone's level 50 or 60 mon early in the game. You will deal with the disobedience problem, but even attacking every other turn can be enough to Curb-Stomp Battle if it's a good Pokémon.
Far more likely, you'll get a "breeding reject"- a level 1 leftover Pokémon from someone's breeding project. These Pokemon usually have good Natures, perfect IVs in most of its key stats, great egg moves, hidden abilities, and an experience point bonus. These mons are excellent for the main campaign and may remain useful when you want to start your own breeding project.
If you were willing to shell out the extra cash for Pokémon Battle Revolution and could spend five minutes finding the proper codes on the internet, you could get your hands on two free Mystery Gift Pokémon for any DS game: Electivire and Magmortar. Both came stacked at level 50 with great physical and special stats (respectively), in addition to some hard-to-get moves that between the two, covered pretty much every type in the game. The only requirements? Have a Pokedex, be near a Poké Mart, and have access to Mystery Gift, all easily done through less than an hour of playing and some more net-surfing. Oh, and, have at least four Gym Badges, or they will disobey you. Together, these two could carry you to the Elite Four, no sweat. And if you actually did some work to win at the Wii game? Say hello to surfing Pikachu!
Inver, the Inverse battle specialist on Route 18, gives out a daily reward whenever you defeat him. The more "super-effective" (that is to say, what would normally be non-effective) attacks you pull on him, the better the prizes. Of course, since Normal type is a Game Breaker there, one could simply spam False Swipe over and over again instead.
The Poke Bank has completely trivialized trading pokemon. You can load up all your pokemon from Pokémon Black 2 and White 2 and transport them into Pokémon X and Y, and your pokemon from both games into Omega Ruby and Alpha Sapphire, letting you start your adventure with all the freshly hatched shiny pokemon you spent so much time breeding, or the level 10 Tornadus, Landorus, and Thunderus you can get in the Pokemon Dream Radar game. In the past, you'd at least need to capture a few common mons first just to have trade fodder to get a few of your pokemon over. That, and the trading process itself, took some time to accomplish. Starting the game with every pokemon already caught, and the Oval Charm in your possession as soon as you get your first gym badge makes things a lot easier.