Part of the idea behind Milestone leveling in D&D is to decrease metagaming and to simplify the leveling mechanic by eliminating XP bookkeeping. The point of character-specific milestones is to engage players more into the world and to reward players who create interesting stories not with monetary or level rewards, but by tying their power progress to elements in their stories. This can also give the table plausible rationales to speed up or delay character level progression as they please. These are less "you level up every time you do these things" and more of "It's time to level up our characters (or a specific character), the DM will assign these tasks as triggers for the level up."

Barbarian Example Milestones:

defeating specific powerful foes in single combat

advancing spiritual or religious goals

advancing tribal goals

using your latest acquired class feature to aid in resolving a challenging encounter

Bard Example Milestones:

Learning new notes or words of power by discovering lost secrets

Learning magical secrets through personal research into music and performance

Discovering new martial or magical secrets from hermits or exotic monsters

Honing skill expertise through legendary exploits (using the involved skill to defeat a challenging encounter)

Cleric Example Milestones:

Completing major quests for your deity

Establishing worship in a new locale

Resolving a major character arc that brings your character closer to your deity

Sacrificing significant other goals or life priorities for the sake of your deity

Druid Example Milestones:

Spending an entire adventure in the company of multiple beasts

Resolving a challenging encounter in favor of beasts or nature

Spending extended time in Wildshape or in the land holding mysterious rites

Exploring the Feywild or Elemental Planes and testing your mettle against its denizens; performing your rites in Feywild or Elemental Plane nodes

Fighter Example Milestones:

Finding a more skilled Fighter who knows the skills you seek and spending Downtime learning under their tutelage

Testing personal theory and practice in battle and successfully hitting with disadvantage using provisional new class abilities

Spending time using new skills to overcome challenging encounters (calligraphy for the win)

Learning new abilities by researching and training with a friendly Wizard or Druid or Cleric (or Giant or Psion)

Monk Example Milestones:

Unlocking access to special training with higher ranked Monks by completing important quests for your temple

Acquiring new insight by spending time meditating and overcoming challenges in the appropriate environment (Shadowfell, elemental planes, etc.)

Building up resistance to poison by incrementally taking more and more specially prepared poison every day

Socially Interacting with at least 15 different languages

Paladin Example Milestones:

Officially swearing an oath at a holy place or among a significant brotherhood

Training with a higher level Fighter who has the skill you desire

Facing a horrific monster without giving in to fear

Completing a challenging goal dictated by your Oath

Ranger Example Milestones:

Hunting down a new Favored Enemy of a significant challenge

Spending time with Beasts or in the environment appropriate to the class feature

Extensive overland travel in the wild

Hunting down Legendary monsters in your Favored Terrain

Rogue Example Milestones:

Breaching 1000 checks (success or fail) with a specific pair of skills

Studying anatomy at a prestigious college for healers (or assassins)

Surviving 100 area damage effects that challenge Dexterity

Using Skills to overcome a challenging encounter

Sorcerer Example Milestones:

Uncovering insight to your true nature in lost tomes or secret places

Defeat a challenging encounter relevant to your Origin

Stumble onto a reflexive talent while supposedly researching spells in a wizard academy

Undergo apotheosis by acquiring a powerful artifact or completing a major quest related to your Origin

Warlock Example Milestones:

Complete a major goal advancing the interests of your Patron

Discover forbidden secrets allowing you to extract even more power from your Patron without their consent

Trick your Patron into a better deal directly

Acquire more insight into your Patron by completing quests dealing with their lore and nature

Wizard Example Milestones:

Acquire and study a legendary spellbook containing powerful spells

Train at a world-reknowned Academy using extended Downtime

Personal research and experimentation using raw insight learned from broken sigils and abandoned laboratories

Wrest knowledge from the gods from a site protected by a powerful guardian

A level or two doesn't quite matter in the balance of things, especially if you're in a relatively large party, so slightly delaying (or advancing) leveling until the players can play out the relevant personal milestone, especially when intimately connected to their personal narratives can make it seem more alive and engaging. This can also be limited to specific class abilities to make it easier for the DM - that is, the character gets other leveling features at the adventure milestone, but the specific class feature isn't acquired until the character completes the specific personal milestone.