Here's how I would describe the gameplay loop during missions:
1) Check the objectives, listed on the right of the screen: Is the mission timed? Are there time extensions? Is there a kill count goal? Outposts to take? An allied base to defend? NPCs to escort? Is it a race to the finish? Or something else (i.e. collect X number of rupees, find specific treasure, etc)?
2) Check the map: What's the most efficient route to target? Are my allies together or separate? Is each route already open, or are there gates to open by defeating enemies with yellow circles on the map? (To examine map more closely: Pause with +, then tab left with L, then switch to map with L3 and zoom with R3.)
3) Fuel up: Plow through mobs of enemies as you run to your destination. Juggle bodies (even defeated ones that have turned black) to fuel your special attack meter.
4) Crates, barrels, etc: Smash them for rupees, materials, apples, sometimes elemental ammo. Wizzrobes and some other elemental enemies also provide ammo. (The challenge "A Feast for Wizzrobes" is brilliant for farming these.)
5) Strong enemies - the core of the gameplay: It's all about getting the weakpoint gauge to appear and then efficiently depleting it so you can execute a weakpoint attack with X for massive damage. The gauge can appear in a number of ways. The gauge appears after the enemy performs certain attacks, or after you perform a "perfect dodge" at the last second -- the latter triggers slow motion and prompts you to execute a "flurry attack" with Y, rapidly depleting the weakpoint gauge. Alternatively, when the enemy winds up for a big attack, a rune sometimes appears over their head -- use the corresponding rune (with proper aim and timing and spacing) to counter, stunning them and revealing the gauge. Special attacks deal damage and also reveal the gauge, but only for a short period of time -- however, it's still long enough to finish off the gauge if the gauge is low. Elemental rods immobilize the enemy and reveal the gauge for an extended period of time (but enemies of the same element are immune). Other ways to reveal and deplete the gauge: Bombs chip away at the gauge. Stasis prolongs it. Strikes to the face (i.e. arrows) stun and reveal the gauge. Parries (ZL to guard + Y to reflect attacks) reveal the gauge. Some aerial attacks reveal the gauge, such as Mipha's ZR waterspout warp. Explosive barrels, cryonis in pools of water, lightning in water or on metal crates, and other environmental hazards reveal the gauge.
So if time is tight and the weakpoint gauge is full, use rods to significantly damage the weakpoint gauge, special attacks to polish it off, and then rely on perfect dodges and counters. Try to lure strong enemies close together so multiple enemies get hit by the elemental rods, special attacks, weakpoint attacks, bomb blasts, etc. Also be mindful of incoming attacks from other enemies while locked onto a target -- you can even perfect-dodge one enemy's attack to attack another enemy in slow motion.
6) And finally... Watch your health! Eat those apples! Keep an eye on the clock! Switch to your allies to save keeps! Don't forget to blink! As the game progresses, elemental variants appear for every enemy, and the mobs lean more heavily on projectile spam, kamikaze attacks, etc, so it can get crazy. :)