Introduction Defensive Partners
Skarmory might be the best partner for Lanturn in general. Purely based on typing, these two form an excellent defensive core. Lanturn’s weaknesses are Ground and Grass, one of which Skarmory has an immunity to, the other it has a 4x resistance to. Skarmory’s weaknesses are Fire and Electric, one of which Lanturn resists 2x, the other Lanturn gains HP from. Skarmory’s high Defense stat helps cover one of Lanturn’s largest flaws, its physical defense. Skarmory, whenever it comes in, lays down Spikes, which Lanturn benefits from to help break down teams. The mere presence of a Lanturn is sure to make your opponent think twice about clicking Thunderbolt with their Electric type, like a Zapdos or a Jolteon, or a Pokemon reliant on boltbeam coverage such as Gengar. Lanturn also annoys one of Skarmory’s biggest enemies, Claydol, as no Claydol wants to stay in and get pounded by a Hydro Pump. Another Pokemon that can come in and exploit Skarmory is Cloyster, which Lanturn quickly dispatches with a Thunderbolt. 
Flygon does not have to use a defensive set to pair well defensively with Lanturn. Flygon greatly appreciates Lanturn’s ability to absorb ice type moves from faster threats like Starmie or Gengar, while subsequently forcing them out. Lanturn benefits from Flygon’s ability to pivot into Edgequake coverage with minimal concern. If an opponent’s primary way of removing Lanturn is with Dugtrio, Flygon can exploit the Dugtrio user by coming in off of a choice locked Earthquake and set up a free Substitute, or fire off a strong attack. Flygon’s immunity to spikes allows it to switch in with good frequency, as opposed to a Steel type Rock resist. Lanturn can shore up some of Flygon’s weak points due to its ability to wall and threaten non-Swampert Water types. Similarly, its 4x steel resist helps with Flygon’s Metagross struggles, forcing Earthquakes that Flygon can switch in on. The prediction that comes with this switching can be alleviated by running Protect as your last move on Lanturn, thus allowing far safer pivots between Flygon and Lanturn.
Moltres appreciates Lanturn defensively more than Lanturn appreciates Moltres defensively, but this works regardless. The biggest offensive deterrents to Moltres specially all have trouble with Lanturn. Those would be: Starmie, Suicune, Timid Zapdos, and Jolteon. The latter two Pokemon can hit Lanturn with an HP Grass, but they have to play a guessing game in front of Moltres. Alternatively, they can use Thunder Wave to cover both staying in and switching to Lanturn, at the risk of taking a Moltres hit, meaning that this is by no means a rock-solid defensive core, but still worth mentioning. There also is the issue of this core being very frail physically (outside of switching into Heracross, which Lanturn appreciates) as both Pokemon are unable to take Rock Slides well, so this core necessitates a good Rock resistance. If using a Steel type rock resist, Protect in the last slot for Moltres will greatly ease prediction if you happen to come face to face with an Aerodactyl.Some other noteworthy Pokemon include: Porygon2, who reverse traps Dugtrio and allows you to deal with Salamence more easily (even though Lanturn beats mixmence), the Steel type Rock resists, such as Metagross or Jirachi (the latter of which can pass wishes to sustain Lanturn), and Rapid Spin users, who can help alleviate Lanturn’s spikes troubles, specifically Claydol and Forretress. There is also some merit to Cloyster, which switches into Swampert effectively while Lanturn deals very well with Starmie, a giant nuisance of Cloyster. However, Skarmory is my spiker of choice.
Offensive Partners
Lanturn functions similarly to Jolteon, in that it is a solid abuser of Spikes. Teams that are overly reliant on Snorlax as their switch-in to Zapdos or Jolteon will struggle with Lanturn as well, as the Spikes damage racks up very quickly, and the Hydro Pumps don’t make that any easier. The way Lanturn differentiates itself from Jolteon offensively is the ability to kill Tyranitar and Claydol with very little chip damage (anywhere from 1 spike to 20% for Tyranitar, 20% for Claydol). Hydro Pump/Surf is also much more spammable as a STAB move as there are no immunities to it, and the three types which resist it (Grass, Water, Dragon) do not appreciate Lanturn’s other moves, Ice Beam and Thunderbolt. This, of course, comes at the cost of speed, so it is up to you to decide which one you want to use.
Specifically, Beat Up Dugtrio. Lanturn relies on coverage, not power to make progress in a game. Dugtrio is a great partner because it successfully traps most of the Pokemon Lanturn just doesn't have stats to muscle through. Pokemon like Blissey, Jirachi, and Celebi, all of whom can shrug off and set up on anything Lanturn can throw at it, are all baited in with Lanturn and promptly removed by Dugtrio. This pairing allows Lanturn to work through its three biggest obstacles, allowing its unique traits to shine. For example, a team composed of Tar/Blissey/Pert/Gengar/Skarm/Starmie will begin to shudder at the thought of switching into Lanturn with Blissey removed. Moltres destroys two of the aforementioned biggest roadblocks to Lanturn, Celebi and Jirachi (this is not as true if you are fighting a SuperBi or SuperRachi, as those pose an actual threat to Moltres). If you are running a Dugtrio alongside both Lanturn and Moltres, it also greatly benefits from the removal of Blissey and Tyranitar.
Moltres, but it does not care as much about Blissey/Snorlax/Tyranitar thanks to Focus Punch, at the cost of some special power. Also has a slightly better speed tier. 
Breloom is a great fit on Lanturn Spikes teams, as it forces switches, racking up plenty of Spikes damage. Using Breloom makes you more susceptible to Dugtrio, however, with Spikes down, Dugtrio cannot switch in many times before finding itself in Mach Punch range. Breloom obviously pressures Pokemon like Tyranitar and Blissey, the former of which physical variants Lanturn struggles to switch into, and the latter definitely does not appreciate fighting moves. Breloom also has some decent defensive synergy, as it resists Edgequake coverage, in a similar vein to Flygon, while also resisting Grass moves. And, as we have seen before, Breloom + Charizard is a tried and true combination, why not try a Lanturn with it?
Specifically mixed Tyranitar. Slamming Blissey/Snorlax with a Focus Punch/chipping away at it with Brick Breaks while eating away at the rest of your opponent’s team with Tyranitar’s excellent coverage makes Lanturn’s job much easier. Lanturn also is a beneficiary of Sand offensively, as it allows it to break down Snorlax after a couple switch-ins with Spikes down. Lanturn can come in on Pokemon that would otherwise threaten more offensive teams with Mixtars such as Starmie, and pick up where it left off. This set in particular I have found great success with on my Lanturn teams.
Specifically mixed Salamence. Very much in a similar vein to Tyranitar, you rip holes in your opponent’s team with its great coverage, and cover its defensive shortcomings against Pokemon such as Gengar or Starmie with Lanturn. Brick Break doesn’t hit as hard coming off of Salamence, and it certainly is not Focus Punch, but it does the job pretty well.
Specifically mixed Zapdos. Drill Peck allows you to break down Celebi, meaning it has to be very careful when switching into Lanturn later, as it can fall to an Ice Beam. Running a very physically oriented Zapdos with HP Fighting can allow you to bait Blissey/Snorlax as well. If you are considering using HP Fighting, I would strongly recommend using HP Grass as the 4th move on Lanturn to better deal with Swampert. If not running mixed, Baton Pass with a Dugtrio in the back is almost a necessity (unless you can find other ways to exploit Blissey/Celebi), as otherwise Blissey and Celebi will effortlessly ruin you. 
Specifically Focus Punch Gengar. Another way of baiting Blissey/Lax, and if you opt for Explosion over Substitute, it’s a great way to remove a threat to the rest of your team (albeit prediction reliant), while providing solid speed and spin blocking.
One of the biggest things you have to consider in team building with Lanturn is how to make up for the speed you lose by not using Jolteon. There are a few ways of going about this. You can either run naturally fast physical attackers such as Aerodactyl, or run some other ways of making up speed. This includes, but is not limited to: Agility Metagross, Salac Berry Metagross, Salac Berry Flygon, and Dragon Dance Salamence. On the special side, consider running Agility Moltres or Agility Zapdos.Here are a few fun replays (not all by me):
https://replay.pokemonshowdown.com/gen3ou-1162297556
https://replay.pokemonshowdown.com/gen3ou-1165054118
https://replay.pokemonshowdown.com/gen3ou-1171162345-5jsmslezfbg45ii1nifvcapdfg3olhepw
Conclusion shoutouts to nchmax GreenGogoatttt and Hclat for assisting me with writing this I apologize if I'm forgetting anybody else