Exeggcute has a 10% encounter rate in Poni Wilds and Ancient Poni Path, and a 40% encounter rate on Exeggutor Island. Exeggutor is a guaranteed encounter on Exeggutor Island.
Exeggutor might look ridiculous, but that just makes it even funnier when it kicks butt. Exeggutor is pretty decent, even if it's literally the last Grass type and the second to last Dragon type (in Moon in Sun its dead last) you will encounter in the main story. It can be a pretty helpful mixed attacker and can take out some key pokemon effortlessly. If you have a team slot, give Exeggutor a chance.
Hapu (Vast Poni Canyon, Ground-type): Have a field day. Dugtrio can only 3HKO at best, and Flygon can only 2HKO at best. Mudsdale and Gastrodon get OHKOd by Grass Knot, and Flygon falls easily to Dragon Hammer. Dugtrio can take some time to whittle down with Grass Knot, but isn't a threat outside of Sandstorm. Be careful of its Sucker Punch if it can weaken you a lot.
Lusamine (Ultra Space): Avoid Clefable if you don't use Acid Downpour. The best thing Lilligant can do is annoy you with Teeter Dance. Kill it with STAB Dragon moves or super effective coverage, Dragon Hammer can usually kill Mismagius before it can kill you, but be careful of Pain Split and Special Attack drops. Milotic is pretty easy, so kill it with Grass Knot or Wood Hammer. Physical Exeggutor should avoid Bewear, but special ones can kill it with Leaf Storm or Draco Meteor.
Solgaleo (Altar of the Sunne, Sun only): Solgaleo is too good for the poor plant. Just use the Master Ball or something else.
Lunala (Altar of the Moone, Moon only): Lunala is also too good for the poor plant.
Gladion (Mount Lanakila): Avoid Crobat and Weavile for obvious reasons. Fire Silvally can kill you with X-Scissor before you can kill it, but Grass and Water Silvally are fair game with Flamethrower and Grass STAB of choice respectively. You can kill Lucario with Flamethrower as well.
Hau (Mount Lanakila): Dragon/Wood Hammer the Raichu, Dragon STAB or Rock Slide will kill the Fire types, but be careful or Incineroar's Z-Move. Flamethrower or Dragon STAB will take out the grass types, and Vaporeon is taken out by grass STAB. Komala falls to your STABs. Avoid Primarina because it outspeeds and Moonblast has a chance to OHKO.
Elite Four Hala (Pokémon League, Fighting-type): Hariyama only 2HKOs with Close Combat so Grass Knot or Dragon Hammer it. Primeape and Crabominable have Outrage and Ice Hammer respectively, so avoid them. Bewear is only possible for special Exeggutor, as Fluffy walls the physical variants. Poliwrath can be killed with any grass STAB.
Elite Four Olivia (Pokémon League, Rock-type): Relicanth falls to grass STAB. Carbink can only be OHKOed by Leaf Storm, but can't be taken on if you don't want the attack drops or don't have it. Golem falls to Grass Knot, but be careful of Steamroller and crits. Probopass is your only unfavorable matchup. It has Power Gem for solid damage, can paralyze you and can set up sandstorm. Lycanroc is fine to take on, but make sure your at full health to take the Continental Crush.
Elite Four Acerola (Pokémon League, Ghost-type): Sableye can easily be taken out by your STABs. Draco Meteor and Dragon Hammer is the only thing that can kill Drifblim in 2 turns guaranteed and it can pass Amnesia's, Focus Energies, and Ominous Wind boosts with Baton Pass. Using something else is preferred. Dhelmise can be easily taken out by your dragon STAB. Avoid Froslass for obvious reasons, and grass STAB the Palossand to oblivion.
Elite Four Kahili (Pokémon League, Flying-type): The best thing you can do is Flamethrower the Skarmory which doesn't have Flying STAB for some reason. Sit this one out.
Professor Kukui (Pokémon League): Use your Grass STAB to take care of the Lycanroc. Avoid Ninetales and Braviary because of their super effective STABs. You can try to take on Magnezone with Flamethrower, but Mirror Coat will kill you if it survives a Flamethrower, which it will thanks to Sturdy. Snorlax can be taken on with Dragon Hammer or Wood Hammer. Finally all three starters have super effective coverage against you so sit out against his ace.
Tapu Koko (Ruins of Conflict): You resist all of its attacks so if you stay at a decent health you'll be fine.
Post-Game: Go nuts, but Exeggutor isn't the best for the postgame battles.
An Exeggcute starts with Confusion, Worry Seed, Natural Gift and either Sleep Powder or Solar Beam depending on the level, while Exeggutor starts with Confusion, Psyshock, Egg Bomb, and Wood Hammer. Exeggcute also gets Extrasensory and Bestow, but since you have the Psychic TM you can evolve immediately. After evolution Exeggutor gets Dragon Hammer, which is your main STAB unless you go for Draco Meteor. Exeggutor also gets Leaf Storm at level 47, which can nuke most things, but also drops your Special Attack, so don't have it as your only grass STAB. It also gives you a 195 power Bloom Doom which is nice. Nothing else is worth relearning except Seed Bomb, Sleep Powder (if you caught it at a high level and still have it as an Exeggcute) or Dragon Hammer is you missed them.
TMs give Exeggutor crazy coverage. Return, Psychic, Brick Break, Flamethrower, Sludge Bomb, Energy Ball, Grass Knot, and Draco Meteor (by tutor) make amazing coverage and have your main STABs. It can get Sunny Day, Light Screen, and Reflect for team support, and can set up with Swords Dance and Substitute. This things movepool is massive even if Earthquake and Trick Room are post game.
In terms of Z-Moves, Exeggutor has Bloom Doom and Devastating Drake for STAB. Shattered Psychic, Inferno Overdrive, and Acid Downpour are some decently strong coverage moves. Z-Reflect and Z-Light Screen can also help patch your defenses.
Fast Pokemon: Exeggutor is never going first. Some more powerful pokemon you'll want to just kill before they can move, and Exeggutor will not be doing that with 45 speed. Something fast is nessicary to run alongside Exeggutor.
Good Pokémon that fit this description include, among others: Raichu, Persian, Alakazam, Gengar, Crobat, Dugtrio, Ribombee, Lycanroc, Talonflame, Archeops, Minior, Weavile, Scolipede
Steel Types: They resist and can kill all of Exeggutor's weaknesses. Exeggutor can take on Ground types with Grass STAB, they can beat Fighting types with Psychic, and can beat the average Fire types.
Good Pokémon that fit this description include, among others: Magnezone, Dugtrio, Skarmory, Metagross, Togedemaru, Sandslash, Klefki, Klingklang, Aegislash