The Dungeon Master's Guide suggests two systems when it comes to gaining levels and developing characters. There are experience points and there are milestones.
The old XP model is a classic and I think it comes naturally to most players. Defeat a monster, save the prince or disable a trap and gain an amount of XP that corresponds to the difficulty of the task. Gain enough and you will eventually get over the next level threshold. However, I think that 5th edition’s worst rule is the encounter building and XP budget system. In it, the character’s levels define a set number of XP to gain each day. It scales according to monster challenge rating and number of individuals in an encounter. Trying to figure out a well-balanced fight is a chore which involves cross-referencing two or more tables.
Furthermore, the XP system in a table RPG poses several questions. For example, what exactly happens when you gain a level and how can you reach such insights that you are suddenly twice as powerful and equipped with a new set of abilities in the middle of a fight? What about that last, killing hit taught you all of that? Where/when do players gain levels, how does experience work on a psychological level and why are players rewarded when going on killing sprees?
Thankfully, the Hoard of the Dragon Queen campaign solves some of these issues by being written over several episodes, each a milestone in itself. Me and my group assumes that characters reflect on their experiences, talk to other characters and evolve their skillsets outside the game. This way, we start (or end) each session by levelling all the characters at the same time so that each player gets some DM time to talk about their character and new skills, not to mention that I get the opportunity to take notes.
As a side effect, I’ve noticed that my players are not so prone on killing or even engaging every foe they meet. They know that they will get a large reward at the session’s end if they survive and solve the mission(s) as well as smaller inspiration rewards if they play according to their character. As such, it is suddenly profitable to sneak around the Roper, teleport out of harm's way or setting up elaborate hit-and-run tactics.
I like the milestone rule and I think it is great for episodic adventures. The milestone doesn’t have to be the end of a session though, you may treat each successful return to the tavern a milestone, if it fits you better. Plus, it will also be much easier for you to plan or quickly make up encounters as you will only have to worry about party size and level.
From 5th edition on, I am happy to leave experience points to the digital games.