With a huge thanks to my Clan, Immortalis Noctis (thanks, Blip and Isgu!), here's the notes pulled together that provide you with a quick and dirty path to increasing your Power in Destiny 2. 

Complete The Story Campaign

• While levelling spend glimmer as soon as vendors are able to sell you a good upgrade.
• Completing the campaign should leave your character at level 20, and around 200 Power.

200-265 Power level

• Complete Cayde quest for Patrols.
• Open engrams from Rockstar promotion.
• Choose one Planet to grind on.
• Sticking with one Planet means you get lots of the same token to hand in.
• Save these tokens for higher into the Power level grind before handing them in.
• Make sure the Planet isn’t the one where the Flashpoint that week is held at: save that one for higher light level.

Activities To Complete

• Patrols.
• Heroic Public events.
• Adventure missions - not worth doing for grinding as other things are more value for time.
• Some of the world quest (Do not finish as the end reward should be saved for 265+).
• Lost sectors.

Other Notes

• Don’t infuse items: they are a waste of materials that are needed later on. However, if you need to infuse to reach 265 then do it.
• Open Legendary/Exotic Engrams as you get them as they will not level up once acquired.
• Don’t pick up Luminous Engrams until later on as they will level up until you acquire them from the NPC (don't pick up any Quest reward Engram until you hit a watch). 

265+

• Spend Glimmer at the Gunsmith.
• When above 280 power more options at gunsmith should become available to upgrade mods, and get the +5 power mods.
• World quests
• Flashpoints
• Milestones which drop Powerful Gear i would honestly save this for about 280 if possible
• Exotic quests for weapons
  • Use these items to infuse with Legendaries
  • This includes doing the Nightfall for the Rat King gun (do the Prestige version of Nightfall at around 295 Power)

Once all Powerful rewards and exotic quests/engrams have been acquired, create a new character. These quests are weekly but character specific.

• Level to cap and transfer your gear from the first character.
• Some items are class specific, so some may not be transferable.
• Do some Heroic Public events to get better class specific gear before beginning better quests.
• Repeat the 265+ stages above.

At the higher stages (280+) gear will drop with a lower increase of Power and could show a lower Power than your current gear, and still be an upgrade

This is because your gear (Exotics will have a 5 mod, Legendary 3) should have a mod which increases the gear by 5. When infusing gear it is the base Power that is used, so the power without that mod. (Example: 280 with a +5 mod will show as 285 and so will upgrade if infused with a 284 power item to 289 (284+5 mod))

Once your three characters have all of their quests completed and all gear acquired from quests, it's best to go back to farming heroic public events. Progress from now will be very slow with Raids recommending 300 power level, something that is easily attainable