Strategically-talking, I maximize my sweeping power since Stallers are a pain to train in-game because of low-leveled wild Pokemon giving few exp and not even being OHKO'd because of common staller's low ATK/SPA. Usually my team consists of 4-5 sweepers, plus 1-2 annoyers that are usually lower-leveled than sweepers. I've never experienced competitive-battling but I think I'd make a poor challenger.
Before I knew that it was totally noobish, I used to combo Swagger and Double team, which give satisfying results if you are a sadistic person.
Before I knew that it was totally noobish, I used to overuse 100%-accuracy state moves such as Spore or Confuse Ray. Also very satisfying it was.
In-game, I pay attention to the game's background and try to do side quests, finding special objects, eventually catching Legendaries. I like to build my team around high type-coverage and I've found fav mons representing each type.
The type I don't play a lot is Dark because in Gen I, II, III (the ones I've played trough) Dark moves aren't that powerful. Also I don't play that much Bug Pokemon, it's somehow a weird type to play in my opinion. I guess Bug's condition significantly have gone up since the releases of gen IV, V, VI new moves... When I can I'll give them a try. Also I never use Dragons because I think they are very mainstream. I admit that not-liking mainstreams things is itself a hype trend that I'm trapped in, and I totally assume that paradox. I occasionally use Normal typed Pokemon but I don't really know how to play them so they have a comfortable spot in my team. I have a preference for Psychic, Fire, Ground, Ice, Water, Grass, Fighting, Ghost, Flying, Electric, eventually Poison moves.
I have one or two well-chosen HM Slaves, sometimes HM Slave is even used for Fly since all my parties don't end up with a friggin' bird. I use Sentret and Pelipper for good HM-slaving, but if you have suggestions I'm interested =D Those caves where you have to use the Flash-Rock Smash-Strengh-Surf-Waterfall combo make me sick really. It tears apart my 6-mon team-using and I don't want to teach HMs to Pokemon in my team, there all all sorts of more interesting moves (except for Surf, Waterfall, Dive, but having these three on one single Pokemon is useless to me).
I do not have much interest in Legendaries because usually they're catchable after the main Quest, which is to beat E4 (i.e. Mewtwo, Lugia/Ho-oh,...). Legendaries are overpowered and do not train as regular Pokemon because of high Exp. Curve. I had any ways fun playing Mewtwo, Lugia, Articuno (OHKO is very satisfying). These three are my personal Top 3. I did not use Gen. III Legendaries in serious battles but I think Kyogre would do good. I've had the occasion to try Shaymin Sky in Pokemon Flora Sky and that was great because I love grass type.
The main issue I encounter with Pokemon games is that you can carry only 6 mons, all the rest is stocked in a PC and when you want to change a partner it's very difficult to grind it up to catch missed levels. I'm not very patient. A good answer to this is the Rare Candy cheatcode, I used it in unofficial releases such as my fav rom Hacks. I hate grinding. But I grinded any ways : I feel that universe is disrupted when 1 mon levels up while others not. I feel like I have to maintain all my team to same levels between all trainers so it's annoying sometimes, and when I begin to have levels high enough I just drop this grinding and let my team having 10-levels gaps sometimes. No more.
My overall aims are completing my Pokedex and making a good battler out of every mon I catch, even if there are ones that I don't even want to train, i.e Caterpie, Wurmple, Remoraid, Zubat, baby Pokemons,...
Build different teams in the same game could be nice. But this is screwed up by making TMs work only once :/, any ways I did not succeeded for now in many years of playing, because of some weariness certainly. I'll try that on Pokemon Glazed : this hack seems to have features that make it last eminently longer than official releases.
I've also began some EV training in RSE-likes... Managing this shit is hell. I have a .xls files where I have to write down every EV gained. Also I have to have an eye on IVs (rare-candied to lv. 50 is easier to get close numbers so yeah). Also I have to check on nature. And I imagine that all competitive-battlers make this road in occurences more numerous than I do. Must really be a hell when you train a whole team this way, but I have to say the results is for sure satisfying (had very good Shayminsky and Lucario with EV-training).
Yea that's a long post I did, sorry, but I find it cool sharing on every aspect on the game :D