Boss Ordering in Mega Man

Ok, there is a lot here so I'll try to be brief:

Mega Man 1:

Fire → Bomb → Guts → Cut → Elec → Ice

Alternate: Guts → Cut → Elec → Ice → Fire → Bomb

Guts Man's stage is sometimes difficult, but his AI is a lot easier on a fresh start, and does less damage. Ice Slasher is also nice to have in Fire Man's stage.

Mega Man 2:

Metal → Air → Crash → Bubble → Heat → Wood → Flash → Quick

Alternate: Metal → Wood → Air → Bubble → Heat → Crash → Flash → Quick

Weaknesses are wonky, as Metal Blade works excellently on Wood, Bubble, Flash, Quick, and Metal (hahahahahahahahaha). I recommend this route as Wood Man's stage is easy enough with Metal Blade, and Leaf Shield is nice to have for Crash Man's stage. Item 2 to take on Heat Man's stage to get Item 1. I like Crash before Flash just to make the stage easier.

Mega Man 3:

Top → Shadow → Spark → Magnet → Hard → Snake → Gemini → Needle

Alternate Magnet → Hard → Top → Shadow → Spark → Snake → Gemini → Needle

There are actually two weakness loops: the first 5 are one loop while the last 3 are a separate loop. I prefer Magnet Man earlier because Magnet Missile and Hard Knuckle are good weapons against large foes, and Top Spin is not so necessary early on. But do remember to abuse its knockback cancel (if you get shot in the air using Top Spin, you don't get knocked back and can continue your jump).

There is a tendency for people to enter the Snake → Gemini → Needle → Snake loop via Snake. However, It is also very possible to enter via Gemini once you know you actually control him. For the first half of the fight while there are two copies of Gemini Man he only shoots when you press B, so you can adjust your jumps to never take a hit. For the second half of the fight Gemini Man jumps when you press B, so you can force him to jump when you want to slide under him, again never taking at hit. With this in mind

Gemini → Needle → Snake

is quite easy and allows you to fly over the final platform section of Snake. Snake Man is a complete joke once you have Needle so if you're having trouble beating snake try this alternate.

Mega Man 4:

Pharaoh → Ring → Dust → Skull → Dive → Drill → Toad → Bright

Pharaoh Man makes a popular opening for an easy stage, access to Balloon, and a multi-directional chargeable weapon. Toad Man and Skull Man are also very easy AI, and popular openings to start the cycle. The only general notes are Ring Man is very mean as a start, and Drill Man's stage is easier with Pharaoh Shot or Skull Barrier.

Mega Man 5:

Star → Gravity → Gyro Man → Crystal Man → Napalm Man → Stone Man → Charge Man → Wave

Generally an accepted pathway, as Star Man's AI is really easy and the barrier is nice for Crystal Man's stage. The AI for the first 8 is one of the easiest in the Classic Games, so there's a lot of variety you can take to this one.

Mega Man 6:

Plant Man → Tomahawk → Yamato → Knight → Centaur → Wind → Flame → Blizzard

Alternate Flame → Knight → Blizzard → Plant Man → Tomahawk → Yamato → Centaur → Wind

The first route is weakness order, while the alternate is divergent. Flame Man is a fairly easy stage with an easy boss. Plant Man is easier, but Flame Man gives you Power Adaptor. Consequently, use it to take out Knight Man (whose shield is worthless against it) and also not have to revisit the stage. Knight Crusher is excellent against large enemies, including importantly the apes in Plant Man's stage. So this route prioritizes a much better arsenal for a later gain of the Jet Adaptor.

Mega Man 7:

Cloud Man → Junk → Freeze → Burst → Slash → Spring → Shade → Turbo

It should be noted that Burst Man and Slash Man are BOTH weak against BOTH Freeze Cracker and Scorch Wheel. Slash Man being the only one with a weakness from the first four means an obvious opening, while Cloud Man is a very easy stage and boss for a very versatile weapon.

Mega Man 8:

Tengu → Clown → Grenade → Frost → Aqua → Sword → Search → Astro

There are two separate loops here, unlike with MM7. The first four is one loop, the second four is a different loop. Tengu Man has a very basic stage and has a fairly easy pattern. Tornado Hold is awkward for some people but is an incredible source of damage alongside your Mega Buster. Aqua Man is the easiest of the second loop, but Astro Man is also very easy if you use Tornado Hold.

Mega Man 9:

Galaxy → Jewel → Plug → Tornado → Magma → Hornet → Splash → Concrete

Alternate - Splash → Concrete → Galaxy → Jewel → Plug → Tornado → Magma → Hornet

I sometimes start with Hornet first but you need to be quick on your feet to take him on first. It would make Splash Woman that much easier, though. Splash Woman gives Laser Trident, which is enormously useful to have in all of the stages, so it is very valuable to get early. Splash Woman is also not too difficult, though not nearly as easy as Galaxy Man is.

Mega Man 10:

Sheep → Pump → Solar → Chill → Nitro → Commando → Blade → Strike

Mega Man 11:

Block → Acid → Impact → Bounce → Fuse → Tundra → Torch → Blast

Mega Man & Bass:

Cold → Burner → Pirate → Ground → Tengu → Magic → Astro → Dynamo

Cold Man is one of the easiest, but with a weakness becomes the third easiest in the entire franchise after Metal Man and Wood Man (who each can get 1-hit-KO'd). This is a standard weakness-order approach, and also gets Remote Mine early which is a good weapon.

Mega Man: Dr. Wily's Revenge

Elec → Ice → Fire → Cut then Flash → Quick → Bubble → Heat

Mega Man: Rockman World II

Metal → Air → Crash → Wood → Needle→ Magnet → Hard → Top

Mega Man: Rockman World V

Mercury → Neptune → Mars → Venus → Pluto → Jupiter → Saturn → Uranus

The weakness order is actually exceptionally wonky and includes the midfight Terra. The order is Neptune > Mars > Venus > Jupiter > Saturn > Mercury > Pluto > Uranus > Terra > Neptune. But the actual access doesn't let you go in this order.

In any case, Mercury gives the Grab Buster which is excellent for lasting through some of the dangerous points in stages, as it heals you. Mercury is also an easy stage with an easy boss. Past that it's primarily following weakness order and substituting the Mega Arm when you don't have the specific weakness.

Here is a list of what order to fight the bosses in each game and with what weapon

Mega Man 1:

Bombman (Use Mega-Buster)
Gutsman (Use Bombs)
Cutman (Use Gutsman Rock Throwing [the two rocks in the room will kill him])
Elecman (Use Cutman Boomerang)
Iceman (Use Elecbeam)
Fireman (Use Iceslasher)

Mega Man 2:

Flashman (Use Mega-Buster)
Quickman (Use Time Stopper)
Metalman (Use Quickman Boomerangs) or (Metal Blades)
Woodman (Use Metal Blades)
Airman (Use Wood Leafs)
Crashman (Use Air Shooter)
Bubbleman (Use Regular Weapon) or (Metal Blades)
Heatman (Use Lead Bubble)

Mega Man 3:

Magnet Man (Use Mega-Buster)
Hard Man (Use Magnet Missiles)
Top Man (Use Hard Knuckle)
Shadow Man (Use Top Spin)
Spark Man (Use Shadow Blade)
Gemini Man (Use Search Snake)
Needle Man (Use Gemini Laser)

Mega Man 4:

Toadman (Use Mega-blaster)
Brightman (Use Rain Flush)
Pharaohman (Use Mega-blaster/Flash Stopper)
Ringman (Use Pharoah Shot/Megablaster)
Dustman (Use Boomering)
Skullman (Use Dust Crusher/Ring Boomerang)
Diveman (Use Dust Crusher/Skull Barrier)
Drillman (Use Dive Missles)

Mega Man 5:

Star Man (Use Mega-Buster)
Gravity Man (Use Star Crash)
Gyro Man (Use Gravity Hold)
Crystal Man (Use Gyro Attack)
Napalm Man (Use Crystal Eye)
Stone Man (Use Napalm Bomb/Mega Buster)
Charge Man (Use Power Stone)
Wave Man (Use Charge Kick)

Mega Man 6:

Flame Man (Use Mega-Buster)
Blizzard Man (Use Flame Blast)
Plant Man (Use Blizzard Attack)
Tomahawk Man (Use Plant Barrier)
Yamato Man (Use Silver Tomahawk)
Knight Man (Use Yamato Spear)
Centaur Man (Use Knight Crush)
Wind Man (Use Centaur Flash)

Mega Man 7:

Burst Man (Use Mega-Buster)
Cloud Man (Use Danger Wrap)
Junk Man (Use Thunder Bolt)
Freeze Man (Use Junk Shield)
Slash Man (Use Freeze Cracker/Scorch Wheel)
Spring Man (Use Slash Claw)
Shade Man (Use Wild Coil)
Turbo Man (Use Noise Crush)

Mega Man 8:

Tengu Man (Use Mega-Buster)
Clown Man (Use Tornado Hold)
Grenade Man (Use Thunder Claw)
Frost Man (Use Flash Bomb)
Aqua Man (Use Mega-Buster/Astro Crush)
Sword Man (Use Water Balloon)
Search Man (Use Flame Sword)
Astro Man (Use Homing Sniper)

Mega Man X:

Chill Penguin (Use X-Buster)
Storm Eagle (Use X-Buster)
Spark Mandrill (Use Shotgun Ice)
Flame Mammoth (Use Storm Tornado)
Armored Armadillo (Use Electric Spark)
Launch Octopus (Use X-Buster/Rolling Shield)
Boomer Kuwanger (Use Homing Torpedo)
Sting Chameleon (Use Boomerang Cutter/Homing Torpedo)

Mega Man X2:

Wheel Gator (Use X-Buster)
Overdrive Ostrich (Use X-Buster)
Wire Sponge (Use Sonic Slicer)
Bubble Crab (Use X-Buster)
Flame Stag (Use Bubble Splash)
Morph Moth (Use Speed Burner)
Magna Centipede (Use Silk Shot)
Crystal Snail (Use Magnet Mine)

Mega Man X3:

Blizzard Buffalo (Use the X-Buster)
Toxic Seahorse (Use Frost Shield)
Tunnel Rhino (Use Toxic Acid)
Volt Catfish (Use Drill Fang)
Crush Crawfish (Use Voltage Gun)
Neon Tiger (Use Crush Claw)
Gravity Beetle (Use Ray Splash)
Blast Hornet (Use Gravity Well)