Starting on Elec Man is a waste of time, since you'll have to do the stage again to get the Magnet Beam, and he's a pain to fight without the Rolling Cutter anyway. Ice Man is equally annoying for new players, with disappearing blocks, Foot Holders, more Foot Holders mixed with a few Pepes (Pengs), Big Eye, Ice Slasher damage... I usually do Guts, Cut, Elec, Ice, Fire, Bomb, although you can also start on Bomb Man first as his stage isn't too hard (spikes aside - I hate the Mambu/Flying Shell section above the spikes). Once you're past those annoying platforms in Guts Man's stage, the game is basically nothing until you get to the Wily stages (specifically, stages 2 and most certainly 4).
On 2 I usually start on Air Man, then Metal, Bubble, Heat, Wood, Crash, Flash, Quick. Alternatively, Crash/Flash/Quick can be done after Metal if you don't mind using Metal Blades on the Pipis instead of the Leaf Shield, or Flash and Quick right after Air Man (the Air Shooter decimates the Sniper Armors found in both stages), and then you have the weakness for Metal as well. In Normal mode, Flash Man dies to the buster in no time; I wouldn't recommend this pattern in the Wily Wars though as Flash is much more difficult on that game (but seeing that you said Legacy Collection, which is basically the NES games in an emulator, the point is moot). Don't start on Heat Man unless you want the game to become Heat Man: The Series.
On 3 I like starting on Magnet or Top, although Snake Man is also fairly easy but I prefer having a few weapons first, and he only takes 1 damage from the buster anyway (Top Spin is also quite useful in this stage on anything that isn't Hammer Joe or Snakeys). I don't recommend starting on Gemini, Shadow or Spark. Probably the best route is Top, Shadow, Needle, Snake, Gemini, Spark, Magnet, Hard. Shadow Blades might only do 2 damage to Needle Man (beats 1 only with the buster), but it's better to get the Rush Jet early IMO - I hate doing Gemini's stage without it, and Shadow Blades just about dominate the entire stage as it is. Just one more thing, don't ever pause the game while you're using the Rush Jet or Marine, as Rush will teleport out and Megaman will drop like someone tied an anvil to him. Make sure you're above solid ground if you have to pause to use an energy tank or whatever.
On 4 I start on Dust Man because he's easy to knock off with the buster alone, and going to Skull next gives you 3 energy tanks (if Eddie gives you anything else, don't pick it up, go down and up the ladder and he'll reappear, hopefully with a better randoly-generated item). So, Dust, Skull, Dive, Drill, Toad, Bright, Pharaoh, Ring. I don't actually recommend Ring Man unless you know his pattern off by heart; he does a ton of damage otherwise and he moves fairly quick. Make sure you find the Wire and Balloon, as they do make the game easier (as with the Rush items) and getting them is 100% completion anyway. Also make sure you shoot Bright Man at least once with the buster before changing to his weakness, as it makes the fight much easier.
On 5 I always start on Star Man. Star, Gravity, Gyro, Crystal, Napalm, Stone, Charge, Wave. Although, Gyro is also easy to start on, and you get the Rush Jet early for your trouble of going there first. Don't forget the 'MEGAMANV' plates in each stage, although the game is still beatable without it (is that an anti-pun?)
On 6 it almost forces you to use the same order, having only three real ways to get through the game without backtracking. Wind Man is easier to fight than Flame Man, and Blizzard's weapon isn't that useful in either Wind or Flame's stage, so start with Wind first (besides, Blizzard hides some goodies in his stage that are only accessible with Flame Blast and/or Power Megaman). I don't even consider going into Plant Man's stage buster-only, it's up there with Ice Man, Heat Man and Gemini Man. Therefore, my boss order is Wind, Flame, Blizzard, Plant, Tomahawk, Yamato, Knight, Centaur. In fact, it's almost in the exact order the bosses are on the stage select screen, except for the top row. Going any other way results in you having to go back to find the Beat plates, as with Megaman 5 (except that there are only four of them, and all of them require the Rush Power/Jet adaptors to get to the bosses holding them, so Flame and Plant have to be done early regardless).