Persona 5

General Stuff

When the child asks about the cat on the train, the correct answer is "...time to Button Mash!".
When you get near the end of the first dungeon, just blitz on through to the treasure room instead of burning a day to rest and come back at the Safe Room nearby. P5 works differently to the older games, and the final Boss is done as a separate run at the Treasure after you send the Calling Card.
Standard Persona/SMT rules still apply; The leader dies, you are done.
The game doesn't actually mention it, but you can swap party members any time you want outside of combat in Palaces/Mementos. Go to the Stats menu and press Square to toggle active party members.
Light and Curse now have actual damage spells (instead of just %-to-instakill spells)
The Temperance and Star Confidants are well worth doing (you'll encounter the first as a story event, the latter is just suggested you should go meet someone at a place later in the game). Temperance will open up a little more free time than you'd have otherwise, albeit at the expense of money initially, and Star gives you some very nice quality-of-life improvements like being able to swap party members during battles.
Learn to use Baton Pass instead of just smacking another/the same enemy with a second round of whatever just works. It buffs damage and healing, so you can either make it hurt even more or pass it to someone with spells to fix your shit.
The book store in Shibuya has a book that improves your stat gain by an extra point from seeing Movies (3 instead of 2).
Go to the movies and watch DVD's. They don't just have descriptions like in the older games, they have audio clips and are fucking magical .
Don't fret too much about being late on returning DVD's. Just return them late at night; like all good night shift employees, he does not give a shit.
Make sure you always have a book on you with unread chapters. Some mornings you can grab a seat on the train, and it's a free chance to read something.
While the night you send the Calling Card will lock you out of most activities with the usual "go to bed" prompts, you can still make Infiltration Tools.
Some party members just plain aren't available as Confidants until after the current dungeon's complete. They're too worried about what's going on, or whatever.
The Moon Arcana Confidant lets you earn XP in combat for your reserve party members.

New to SMT/Persona Games stuff

Buff and Debuff spells exist for a reason. If you run into something with no exploitable weakness, just smack'em with brainwashing, confusion, whatever. Everything aside from bosses are susceptible to Ailments, so go nuts.
You can still, however, cripple bosses with the -unda stat debuff spells, and you'll generally want to to keep them manageable. Dekaja's a spell well worth keeping on-hand to wipe off any buffs they give themselves too.
The early stretch of the game will seem pretty easy once you just bop Shadows with their weaknesses, but this will change pretty quickly to you getting your shit kicked in if you don't keep a handle on doing so. Especially when they start bopping your weaknesses and stack their own One More's.
Never waste a day by just going home to bed. There's always something you can do, even if it's just going to the movies or visiting the nearby public bathhouse.

Persona 5 Royal

There is an ability you get from Ryuji's social link at rank 7 called instakill. It is a passive ability that's the most powerful grinding tool in the game. If you sprint with R2 at an enemy that cons green and collide with them without hitting x or any other engagement tools, you will simply auto win the battle. In Royal, in addition to just giving you a chance at the persona, it also gives you full money/exp, and the win counts as an all-out attack for the benefit of the multiple abilities that give you extra money for finishing the battle as such. The way it is explained leads you to believe that it will trigger at the start of the battle occasionally, and most people do Ryuji very early, and might miss the explanation. It works with both bus form and regular dungeon running, you just need to be sprinting, don't hit X, and collide with a low con mob, and you will have more money than you know what to do with, but even if you don't abuse it, not knowing how it works is stupid.