well, I don't remember playing the GBA games that much, and the original games were too long ago to really remember how i did them, so I'm just going to base this off of DPPt.
For all my runs, I didn't pay attention to nature, IVs, EVs, even abilities. I always go with a full team of 6 around the same level, and usually have some slaves, but i always have someone with surf, and sometimes others will have other HMs. I rarely have a flying type, so I don't usually have anyone with fly on my team. Also, for my movesets, I've never gotten fancy, just always gone with basically the strongest attacks i could get, and aside from a rare move like hypnosis, toxic or confuse ray i always had 4 damage dealing moves.
For the first time I played through, I just wanted to do it without a walkthrough or really much knowledge of the game, with 3 strategy guides and countless times playing through R/B/Y i pretty much knew where everything was, so I was ready for an experience i was unprepared for. I only occasionally would go to a strategy guide, mainly to find when a pokemon would evolve, or to get a map of the cave I was stuck in. (one where you had a partner to heal your pokemon so you couldn't black out, and i could not figure my way out of there. good training though lol).
Second time through I wanted to try something I'd always wanted to do, and that was raise all of my pokemon 10 levels before each gym battle (ie whoever was on my team needed to be at level 10 before the first gym battle, 20 before the second, etc.) Needless to say, it took a really long time, but was really easy to beat anyone. Also only time I raised a pokemon to level 100 without using the cheat for infinite rare candy (it was my Infernape). to determine who i would have on my team, i wanted pokemon that i'd never really used before, but though they were kind of cool.
Third team was my first with platinum, and basically did similar to my first team, except i had a better knowledge of the area and pokemon.
My fourth and latest team that I just beat the elite 4 with about a week ago was made up entirely of eevee-lutions, aside from Leafeon which was in my third team. I had bred a ton of pokemon (not professionally, just to get the species) in the 2nd game (which i traded my 3rd games pokes to) so I traded over my level 1 flareon/jolteon/vaporeon, 2 umbreon/espeon and 7 glaceon to my team to start with. For the most part, it wasn't too difficult.
My next team once I can borrow someone elses DS to trade them over I'm going to have a Dragon team (Nite/Mence/Chomp/Kingdra) along with Charizard and Skarmory (who isn't really a dragon). It'll probably be pretty easy but i've always liked dragons but never trained them with how late in the games they're available.
Oh, and I don't bother playing much after I beat the E4. I've never been the type of guy to defeat games 100% and after becoming champion I don't really see a goal to go for.
Question, though, does anyone try to win contests? I'd probably try if I wasn't terrible at making poffins.