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).
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.