In Mega Man, and most games of its time, jump and shoot is the solution to every problem. If it moves, shoot it. If it doesn't, shoot it anyway. If it's still there, shoot it some more.
Is there a time not to shoot?
There's a decent handful of enemies that cause something bad to happen when shot, but their effects aren't always harmful enough to discourage attacking them. Changkey Maker sits in Mega Man's way until destroyed, and the loss of a light source is not a danger when he's the only source of damage in the area.
Metall Mommy and Sniper Armor leave behind smaller enemies when destroyed, but those are less of a threat than their first form. Pole has the potential to make a room dangerous if a player shoots a lot of eggs at once, but their movement is slow and predictable enough to be a minor concern at best.
Enemies that explode are a good idea, but with the player being able to attack from across the screen, something needs to be done to make Mega Man more likely to be near an explosion.
These two handle it well, with Walking Bomb appearing in staggered terrain, often running into holes that the player can't easily escape from, and Killer Bomb's wave motion makes it harder to hit before it gets close. Killer is used particularly well toward the end of the first game, with Mega Man being pushed towards them by moving water.
Here, we see enemies with more complex behavior. Blocky tosses his three cans at you, Pukapucker's ball bounces forward, and Spring Face Bomb launches its exploding head a good distance away. While the latter two have some trouble actually hitting Mega Man, Blocky goes almost too far in the other direction, leaving only a tiny window of safety between the pieces. The advantage of this is that it gives a reason to think harder about what to hit it with, as Crash Bombs and Atomic Fire can destroy it outright.
Crazy Razy was the first enemy to split apart when shot, and remains one of the best. We have to be accurate to destroy the upper half without causing it to split, and when it does, it can move out of range and follow players through the area. While some variations on this appeared much later, it's a shame the series didn't keep this one around for the Nintendo Entertainment System sequels.
The way Pukapucker's ball moves around to make it harder to hit would have been great on an enemy that was tougher to avoid after splitting.
These are some other examples of enemies that change state or behavior when attacked. Tanishi's faster movement when shot is the main threat, but it has another neat effect in throwing the shell higher underwater, making it much more likely to get in the way. Brain Break becomes more aggressive and works well with the spikes in its first appearance, while Choker Oh attempts to dodge, gaining a boost on its jump if the water above is close enough. While not all that dangerous, giving players a reason to pay attention to the timing of the environment as well as the enemy's reaction is a great idea.
Jet Bomb's appearance brings three distinct threats: its forward movement, the spread of projectiles it creates after hitting a wall, or the four-way split it leaves behind when shot. When one of these shows up, the player has to quickly determine whether attacking will make the situation more or less dangerous, depending on the height of the Bomb and position of walls behind them.
Bomb Flier is another unique idea, lazily moving in a wave pattern until shot, then rocketing forward. This presents both a choice of threats (a slow enemy that takes up more space, or a fast one that is more predictable), and a matter of timing: picking a spot to destroy the cloud that won't result in Mega Man being knocked off a platform.
Kerog and Tama give us a another new reason not to shoot everything immediately. They only create more enemies when all of the others have been destroyed, so it's beneficial to leave one alive until the fight is over. Tama further expands on the idea by giving us two different targets, only switching to the second type when the first is destroyed or leaves the screen.
As a great expansion to the idea behind Changkey Maker, these two work as a pair to provide one of the main stage gimmicks for Bright Man. Shooting 100 Watton causes the room to go dark, while destroying Dompan brightens it again. With gaps to jump in their first section, choosing whether to get rid of Watton becomes meaningful.
An interesting thing about their use at the end of the stage is that the darkness does little to make it harder, but the danger of pits in the first part may cause players to hesitate to shoot Watton, causing themselves more trouble than necessary.
A few Robot Masters jump in response to incoming fire, and taking control of this is key to fighting Crash Man. Big Pets is the best example of retaliation from a boss though, throwing its pieces at you and requiring good timing to make this work in your favor to reach his eyes.
While resource management is always a part of Mega Man games, Boobeam gets special mention for requiring a plan before firing a single Crash Bomb. It was a sensible idea as a stand-alone puzzle, but the less said about how that turned out in practice, the better.
There were also a few cases where destroying an object could backfire, shown in this collapsing room from Mega Man 5 and these breakable blocks with dangerous enemies behind them.
Of all the enemies that punish the player for shooting blindly, I have to pick Sumatran as my favorite. It gives the player time to think when it hasn't been attacked yet, but hits hard and fast after. Its jump carries it above most shots aimed at its sitting position, but a well-placed charged shot can destroy it easily. Its pairings with other enemies later in the game make it more likely for the player to trigger one while attacking something else, and it is more frequently given positions that are hard to attack horizontally, requiring extra thought as to which enemy to attack first, what to hit it with, or where to dodge.