Running a by-the-book 5e hex crawl takes practice. There are a lot of fiddly rules on different pages: you have to skip back and forth between the sections on weather, wandering monsters, getting lost, and random hex contents.
I've been running hex crawls lately and I've boiled down the relevant rules (for me) into a single random encounter chart. Based on the current location/terrain type, the DM fills specific encounters into the chart, Mad Libs-style. The chart does the heavy lifting for determining weather events, chances to get lost, monsters both in their lair and out, surprise, and landscape features.
This chart also reminds me to run a good mix of encounter types: some monsters are friendly! sometimes you run into an inexplicable mystery of the ancient world! Many "encounters" don't lead to combat! (Of the 12 slots on this table, you only need about 5 potential combats.) With a relatively small and varied number of possible encounters, you can design a bunch that you really want to run, instead of lots of "2d6 goblins" filler. The DM, at least, should be excited to roll on the random encounter table. Here's my encounter chart template as a PDF.
Checking for encounters: Roll d6 four times a day: morning travel, afternoon travel, first night watch, second night watch. Any roll of 6 means that you roll on the encounter chart. (Or use the official 5e rule: roll 20, encounter on 18-20. Pretty much the same odds, but I like the traditional d6.)
Rolling on the encounter chart: Roll d12 on this chart while traveling, or d6 while stationary (for instance, while resting). The chart is organized so that stationary encounters can't sneak up on you while you're not moving.
1: Plot advancing creature: This means different things in different campaigns. If you're running a campaign about the rise of Tiamat, you might populate this slot with dragons or Tiamat cultists. In my open-ended game where the characters are pursuing their own goals, I fill this slot with people or groups related to characters, like the drow assassin that's chasing the noble. If you're running a totally plotless hex crawl, fill this slot with a high-level monster (it potentially advances the story by killing the party!)
2: Intelligent creature: Any locale-appropriate group or creature with tool-using intelligence or higher. At night, if the characters hide their camp and don't light a fire, treat this roll as no encounter (unless your intelligent monsters has darkvision or a sharp sense of smell). That's the advantage the PCs get for not lighting a fire.
3: Unintelligent creature: Beasts or unintelligent monsters. Most beasts shy away from fire. If the characters are resting and have a campfire lit, treat this roll as no encounter (unless they're fearless or fire-based beasts). That's the advantage the PCs get for lighting a fire.
4: Ambush creature: Use stealthy creatures or creatures with special movement modes (flying, burrowing, climbing, swimming, incorporeal). All of these creatures can typically take the party by surprise, so check for surprise against the party's Perception (rules for perception while traveling: PHB 182). If the PCs are currently using a special movement mode, populate this slot entirely with matching creatures (flying PCs may ignore almost all other encounters, but a 4 is always another flying creature.)
5: Beneficial creature: There are actually a few good monsters in D&D, along with friendly adventurers, kobold bands looking for a new king, and suspicious traders with valuable information to sell. You could roll d4 on this chart to find out what kind of beneficial encounter this is.
6: Weather: If you make the standard 4 random encounter checks per day, you have about an 8% to 12% daily chance to hit bad weather. (The DMG weather chart gives a 15% daily chance of heavy precipitation. Of course, this is probably lower in practice because few DMs roll on the weather chart every day.) Feel free to use any place- and season-appropriate weather that challenges or inconveniences the characters in some way, or use the official weather rules in the DMG p. 109. Possible weather inconveniences: while exposed to the weather, you can't benefit from a long rest; low visibility forces a Survival check to avoid becoming lost; fords and valleys are flooded.
7. Lair: Locale-appropriate bad guys (or beasts) live here. Usually lairs are where creatures keep their treasure. This could also be a dungeon entrance. No matter the level of the PCs, I make 1 in 6 lairs contain monsters with more than 10 HD/level/CR. Alert PCs shouldn't run into a cave without scouting first.
8. Survival Check or Hazard: The rules for getting lost (DMG 105) are vague: a Survival check is made "when you decide it's appropriate." Consider this encounter slot a reminder. Characters might get lost because of detours, low visibility, or hazards. Hazards include rockslides, quicksand, etc, all detailed in the DMG p. 110.
9. Path Choice: Take a forest shortcut? Ford the river or caulk the wagon? The tradeoff might be apparent (safe path vs. quick path), or a Survival check, or good reasoning, might be needed to reveal which choice is best.
10. Beneficial location: Typically, this means a friendly settlement or homestead (1 in 6 chance of being bigger than a village). Random settlement rules are on DMG 112. In the uncharted wilds, this might instead mean a treasure or natural resource, or a magic resource like a stand of healing herbs or a teleportation circle, or (valuable late in the day) a defensible place to camp.
11. Ruin: One cool thing about the 5e assumptions is that ruins seem to be about as common as civilized spots. A ruin might be a lair or the entrance to a dungeon, but it might just be an abandoned village or castle, an ancient monument (DMG 108), or a weird locale (DMG 109) that hints at lost history beyond the scope of the adventure.
12. Tracks: It's cool when the PCs gather information that lets them make informed decisions about their surroundings. Roll d12 on this table; there are tracks, noises, glimpses, or other signs that lead to (or let the PCs avoid) that encounter or location.
OK, so much for the chart explanation. Now here's an example chart that I've made for my campaign, and some blank ones in case you want to print them up and use them.
ENCOUNTER CHART FOR THE WILD HILLS:
1: Plot advancing creature: Depends on the group. Let's say a monk who's challenging the monk PC to duel for an available position in the heirarchy: someone murdered the Grand Master of Flowers.
2: Intelligent creature: A paladin from a well-known paladin order. He tells the PCs that he has fought through Hell and returned with a book of devil truenames, and he is fleeing from a pack of vengeful devils. He will accept any help: fight his pursuers (encounter 5); escort him to his destination, which is a holy priory; take his book from him for safekeeping; etc. He is actually a blackguard and he is on his way to sell the book to a devil in a ruined priory.
3: Unintelligent creature: Ghosts of an extinct dwarf clan. They snipe at the party with their ghostly flintlock rifles; from their ancient dwarven curses, it appears that they think the PCs are goblins. They are nearly mindless and cannot be reasoned with. When down to 1/2 HP, each starts retreating to a jumble of bones and treasure in a valley about a mile away. If the bones are blessed, the ghosts will rest.
4: Ambush creature: Hungry wyvern family; will try to fly off with the first PC casualty. Target horses preferentially.
5: Beneficial creature: A troop of paladins searching for the blackguard who stole their book of devil truenames. They will bless and heal friendly PCs and will offer a reward for the book's return.
6: Weather: Unseasonal snowstorm which follows and surrounds a pack of 7 ravenous winter wolves. Under moonlight, the wolves turn into 7 cursed and miserably cold brothers.
7. Lair: A small tribe of sheep-raising ogres, unusually well-supplied with wool kilts, led by Queen Morag, a relatively industrious and intelligent ogre. If the party seems too powerful to kill, she'll offer to hire out her warriors as mercenaries (100 GP a day or best offer).
8. Survival Check or Hazard: A miles-wide area of canyons and plateaus. It's easy to get lost or hit a dead end in the canyons, while staying on the plateaus requires crossing the occasional abyss.
9. Path Choice: Entrance to a long tunnel which leads in the general direction of the PCs' travel, but descends. Various side passages lead back to the surface while the main tunnel goes to the underdark.
10. Beneficial location: An empty tower with ominous gargoyles up top. They're actually non-animate stone gargoyles. However all the wood floors are rotten and will collapse under more than #500 weight (characters get saves to avoid falling). The top floor (unless it's damaged) has a weather-stained permanent summoning circle which can be activated to summon an imp who will answer one question per day: the imp can cast Scry to try to answer the question. Before answering each question, the imp will demand the answer to a personal question about the asker's life.
11. Ruin: A sloping round tower, three hundred feet tall, completely solid (no inside space). An outside spiral staircase leads to a thirty-foot-wide platform on top, protected by battlements. On the platform are the signs of many old campfires. This is a safe place to camp (except in lightning storms). The bottom of the tower has a gnawed appearance because local peasants have removed stones for their building projects.
12. Tracks: An unseasonal path of quickly-melting snow which leads to encounter 6.
Here's a blank encounter chart PDF for you to fill out.