Dear Petzbreeder,
When I play a Pokémon game I take it easy and focus on story. I still like to catch every new Pokémon as it comes, but I won't go out of my way fishing for Feebas, if you catch my drift.
The best way to play is to keep pace with a friend who has the companion version, if that is an option. The in-game rival battles are a good checkpoint to link up with your friend for a battle or trade. This is even easier on the newer versions which will communicate over the internet.
If you're playing both versions solo AND you are going for the living Pokédex, I recommend running a mixed field team and focusing on one version. You will want to have one main team shared between both game versions, just to minimize the monotony.
So the basic strategy is this:
1. Play both games independently until you are able to catch Pokémon and trade. Trade so that you have both starters on the main game.
2. On your "main" game, focus mainly on the story, and don't grind unless you need it to advance. Keep one to three Pokémon from your other game on your main team. Make out your boxes and catch/sort any Pokémon you come across; however, unless you want them on your team, don't evolve them for the dex yet.
3. Rival battles are checkpoints. After defeating your rival, trade Pokémon back to the secondary game and catch up in the story, especially the gyms. Just be careful not to trade any Pokémon that need friendship to level up since I think that gets reset upon trading.
4. Once you've cought up to the checkpoint, trade your main team back to the main game and repeat from #2.
Once you are in the endgame, around a third of your living Pokédex should be filled out just from things you catch and sort along the way. It's just a matter of filling in the blanks.
~Max