For the most part, Destiny 2's Forsaken expansion has streamlined the power leveling process. The new level cap is 50, and the new power (or gear) cap is 600, with a soft cap of 500 at which point only 'powerful' gear rewarded from weekly and daily challenges will raise your level. If you want, you can just complete all the powerful challenges available to you each week and gradually level up, but there are a few ways to optimize this process for faster and bigger gains.
Hitting 500 power
Note that your power level is calculated using all the gear you own, not just what you have equipped. So if you have a high-level gun that you don't like, even having it on your character or in the vault will benefit the level of your drops. Or you could infuse it into the gun you actually want to use, which brings us to Forsaken's new economy.
Infusing guns is much more expensive now—it even costs masterwork cores! As a result, you should infuse sparingly, especially while you're leveling. You're going to get new powerful gear frequently, so if you try to infuse the same piece of gear after every drop, you're going to run out of materials quickly. You're better off just wearing the new stuff for a while, even if it makes you look awful. The good news is that you can infuse the same items at a discounted rate. So, if you have a low-level Better Devils with a great roll, and you get a high-level Better Devils with bad rolls, you can infuse your old Better Devils for just glimmer because its the same gun. This can seriously cut down on costs, so try and infuse using duplicates whenever possible.
Start by playing through Forsaken's campaign (and unlocking a new subclass along the way). You should finish around XP level 47. To hit level 50, I recommend finishing the quest to unlock the Dreaming City, as this area will be important later on. Speak to Petra and Spider in the Tangled Shore after the campaign to start this quest. You should be close to 50 by the end of it. If you're a little short, grind some public events. Be sure to keep Fireteam Medallions sold at the Eververse store active whenever possible. These cost 50 bright dust apiece and increase your experience gains.
Once you reach XP level 50, you'll receive rare and legendary drops which will boost you to 500 power very quickly. You can speed up this process by trading in vendor tokens. Once you reach 500 power, you'll need to shift to powerful challenges to increase your level. That also means you shouldn't complete any powerful challenges before reaching 500, as you'll basically be wasting them. This includes completing exotic quests like the one for the Ace of Spades hand cannon.
Powerful rewards and where to find them
Powerful challenges can reward weapons or armor, and depending on what type of gear you need, your reward may not increase your level. For example, if you have strong weapons but weak armor and pull yet another Edge Transit heavy weapon from a challenge, you probably won't gain a level. In the end, it comes down to RNG, but you can tip the scales in your favor.
Crucially, not all powerful drops are equal: some reward bigger jumps in power and they become less potent as your level increases. Different powerful engrams have their own internal soft caps, and once you reach those caps, they will only drop items that are one or two levels above your overall power level. So, first let's group all the powerful challenges according to their potency.
Small power (soft capped at 520)
- Crucible: weekly and daily matches (rank-up rewards are large power)
- Gambit: weekly and daily matches (rank-up rewards are large power)
- Strikes: weekly and daily runs
- Spider's weekly bounty (the one that costs five ghost fragments)
- Heroic Story Missions
- Planetary Flashpoints
- Ikora's weekly challenge
- Hawthorne's clan engrams
Medium power (soft capped at 540)
- Nightfall Strikes: weekly and 100,000-score runs (note: rare blue gear that drops in the Nightfall can drop up to five below your overall level, which can help you fill in gaps if your helmet or boots or what-have-you falls behind, particularly as you can repeat the Nightfall as many times as you can handle).
Large power ( soft capped at 560+)
- Petra's Dreaming City bounties: completing eight daily bounties and the weekly Ascendant Challenge
- Prime Engrams (these are somewhat unpredictable, but always give a significant boost)
- Exotic weapon quests (these can be completed on alternate characters even after you've unlocked the weapon)
There isn't a universally correct order to complete these challenges, but there is a golden rule to follow: if all of your gear is relatively close to your overall power level, go get a medium or large power reward, ideally one of Petra's bounties. This will spike your power using only a few pieces of gear, allowing you to use small power rewards to fill the power gaps in your other gear. Let's look at an example.
Let's say I'm 520 power overall and all of my gear is between 519 and 521 power. I complete Petra's Ascendant Challenge bounty and receive a 528 power weapon. Now I'm 521 power overall. As a result, my small power rewards will drop at 522. If I'd cashed in my small rewards before doing Petra's bounty, my small rewards would've dropped at 521, so I've basically doubled their value. Now I can get a few small rewards to bring my old gear up and increase my power level before my next large drop. If I'd gotten another medium or large drop before filling in those gaps, it wouldn't be as big of a jump, and if I'd gotten another heavy weapon, it wouldn't help me at all.
In other words, if all your gear is close in level, you should get a medium or large drop to give you a boost. If lots of your gear is below your max power level, you should get small drops to even things out. Again, this all comes down to RNG: it's entirely possible to get helmets from all of your large rewards for the week, or to get boots from a small reward right after pulling boots from a large reward. That's just part of the grind, but by chasing large drops with multiple small drops, you can maximize your overall power gains.