Leveling 1-20 requires EXP. Most activities in the game give you EXP - the main story, public events, side story, crucible, patrols, lost sectors, and even aimlessly killing mobs.
If you finish the campaign and talk to the final NPC at the very end, even if you are level 17/18/19, you will be immediately boosted to level 20.
Each level 1-20 has a cap in how high Power you can attain in that level; you wont be able to hit 260 Power below level 20.
Level 1-20, in the open world, is scaled to the mobs you are fighting and the public events you do, so you can fight next to a leveled 300 Power player with the same potency at level 5 when it comes to open world mobs.
At level 20, you start focusing on leveling Power. There are a few interesting mechanics around this:
Since Power is the average of all 8 slots of gear you wear, you must always try to focus on and prioritize your weakest slots as you level.
Blue engrams and Blue drops decrypt from anywhere between -2 to +4 Power level from your overall average Power level.
Legendary engrams often decrypt beyond +4 levels from your average Power level, up to a softcap of 265 Power.
You may receive legendary gear that is beyond 265 Power from vendors; that will be mostly likely because it has a +5 attack/defense Legendary mod on the gear already installed. So you may receive up to 270 gear from vendors on rare occasions. You may also receive just the legendary mods on their own that you can insert into your gear to boost its power by +5. The gunsmith drops these legendary mods quite frequently in his rank-up packages.
Remember to open your engrams as soon as you get them - their Power is determined by your Power the moment you acquired them, instead of your Power when you decrypt them.
As a fresh level 20 that just beat the campaign, you are free to do any activity to gain more Power. Your best bet at this point is to staw away certain activities that reward gear past the softcap of 260 for later - specifically, do not turn in milestones, nightfall, exotic weapon questlines, or cayde's treasure maps. Instead, focus on activities that reward blue gear. In order of the fastest to the slowest for attaining blue gear, you can do
Public events: around 1-2 blue pieces, chance of legendary and exotic pieces, glimmer and vendor tokens (20 vendor tokens = 2 legendary pieces). They are by far the fastest if you hop from event to event. However, they can get tedious fairly quickly. Try your best to activate the "Heroic" version of these events as they are more likely to give 2 blue pieces instead of 1 upon successful completion and a potentially higher chance at an exotic.
Farming treasure chests, killing yellow bar elite wandering mobs, looting lost sectors, looting golden chests -- all are amazing for getting vendor tokens, but are slower on average than spamming public events when it comes to getting blues. However, when you want to get purples, do these more frequently instead for vendor tokens.
Side story, vanguard strikes, crucible matches - fun original content but quite slow when it comes to amassing a bunch of blues compared to the methods above.
Most of these activities will provide you enough blues over time to get 260. While doing these, remember to occasionally open up faction packages when your gear plateaus to get guaranteed legendaries that boost your average higher and thus get you better blues.
At 260, the softcap for blue gear, you can choose to progress by amassing legendaries.
Legendaries go to 265, but at times drop at 270 from vendors because they come premodded. You can also add your own mods to your gear to push it to 270.
If your only goal is to get these vendor legendaries, focus more on farming tokens instead of public events. Public events have a chance at dropping exotics, but at 260-270, you're better off progressing via guaranteed vendor legendaries and mods.
After 270, the only way to progress is by the methods you patiently saved for yourself this entire time, namely:
Milestones that reward "Powerful Gear" purple engrams. These are on a weekly lockout.
Exotic weapon quests that reward various exotics that scale beyond 290 light. Do them when your weapon slot is holding you back.
Nightfall drops. Weekly lockout.
Possible Cayde treasure map drops.
Once all of those are dried up, you can choose to progress further by