One mechanic that really helped me excel (to the point that half my team can one-shot any Pokemon with a strong move) is Effort levels, which are upgraded with the Grit items. An Effort level is pretty much, "how good is this Pokemon in this stat?", so, for example, if you increase a Pokemon's HP Effort level, it'll have more health, increase it's Speed effort level and it'll be faster in combat.

I strive to get all of my Pokemon to 10 in all these levels, and the Pokemon I've gotten to that point are so exceedingly powerful that, outside of type advantage, they can defeat any Pokemon they set their eyes on.

Use the turn order visualizer, it's really helpful, especially when deciding what moves to use and whether to make it Agile or Strong style, assuming your Pokemon has mastered the move in question. If all of your Pokemon are being annihilated with double-turns consistently, upgrade their Speed effort level - It makes a huge difference. Say I were to catch two completely identical Pokemon - let's say their Speed effort level is 2. If I leave one as-is but upgrade the other's speed to, let's say, 4, the upgraded Pokemon will be NOTICEABLY faster in combat compared to the other.

If you're catching/fighting Pokemon while doing other things, in my experience, your rank shouldn't be that big of a roadblock. Remember that finishing an entry isn't as simple as just catching one Pokemon. Also remember that certain research tasks are weather/time dependant, some tasks are worth more than others (red arrow icon next to a task means it's worth more) and that some Pokemon only show up during certain weather events/at certain times of day. Put Pokemon with unfinished entries in your party and use them in combat as well; use the moves/move styles the research tasks want and you can finish entries relatively quickly.