EDITED after L7R7K7's comments, which made me do some more research and change my post, which wrongly stated that level 30 could be reached without raid gear. So yeah, spoiler warning XD
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I've seen a lot of players being confused and wondering about the progression when they hit 20. Some think it's weird, some don't understand how gear affects your level, so here's a guide I put together. Please comment with tips, suggestion and/or feedback.
How to get to level 30 from level 20:
First, put together a set of legendary armor that compliments your playstyle. A full set might have you at level 24-25-26.
Then, hunt for materials to upgrade your gear. As you increase your armors defense rating and perks, it's light boost goes up, thereby increasing your level.
A full set of legendary armor might not be enough to get all the way to 29, so complimenting your gear with an exotic gear item helps out with that.
If you have a full legendary + 1 exotic armor item all fully upgraded, if you got all your legendary gear outside a raid, it will strangely enough only bring you to level 29. So what's the deal?
First of all, the advantage of being 30 instead of 29 isn't necessarily noticeable. But, if you REALLY want to, here's the harsh truth:
For your armor to reach 120 points of light (which makes you level 30), all your gear has to have +30 light each. There are only two item types that has this; exotic armor... and legendary armor gotten from raids. Hopefully, if all you want is level 30, you haven't first completely maxed out "normal" legendary before reading this. But if you have, no worries. Just go to a raid seeker group on the forum and get cracking ^_^
Bonus tips: Do you have a lot of crucible or vanguard marks to burn, and enough reputation with vanguard, crucible or a faction to buy legendary gear you don't need? When you dismantle a legendary armor piece (that hasn't been upgraded) you most likely get ascendant materials, which is needed to max out any legendary's upgrades.
Bonus tip # 2: If you have your eyes on the prize and only lack an exotic armor piece, find Agent Xur in the Tower on the weekends. If you don't have enough Strange Coins to buy an exotic armor piece for your class, you might be able to buy an exotic engram for Motes of Light. If the Cryptarch turns it into an item for you class, you're covered. If not, you can either save it for an alt character, or dismantle it for more ascendant materials. Either way, you progress.
Social Bonus Tip: When hunting materials in public spaces, look around to find other players. When you find a chest or material, try to attract their attention by shooting, pointing and dancing before picking up the loot. If they return the favour, your material hunting will go even faster, and be more fun. :)
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Don't forget that you can only be 30 with 3 raid pieces and an exotic or a full set of raid gear.