Best ways to get level up past 21

Start with Patrolling. Do bounties, missions, loot chests, farm materials and join public events while out there. When you want to take a break from that, do Eagle/Viper Vanguard Strikes. Alternate between them as desired. You're most likely going to need to buy your first legendary piece of gear. They don't seem to drop unless you already have one. (Or this could just be RNG hating my guts.) So focus on getting up to Level 2 in Vanguard Reputation, with enough Vanguard Marks to buy a piece of armor from your class's Vanguard representative. (Not the Vanguard armory!) If you've been Patrolling between missions and have the Grimore bonus in effect for locations, you'll most likely have Rare gear with Light drop for you enough to get to Level 21 in an hour or two. Making the jump beyond that is the trick. Note, upgrading armor increases its light level. There will be times you'll have to drop a level to give yourself room to gain a level. Do the Daily as soon as you're able. Take friends along if you can. Do the Weekly when you can. Make darned sure to take friends along. Do as many Vanguard Strikes as you feel up to. Take an Exotic bounty when offered. You'll have a choice of three. Look them up before selecting. Some are more difficult than others. Keep an eye out for the Queen of the Reef events. The Legendary gear she passes out isn't the best, but she gives it out like candy. Xur is your friend. Or your corrupter. Hard to say, really. Buy a piece of armor from him when he has one that's appropriate. If you haven't picked up a Legendary weapon somewhere, grabbing one of his exotics isn't a bad idea.