Assassin's Creed Odyssey: Fastest Way to Level Up

Max level in Assassin's Creed Odyssey is Level 50.
To level up quickly, the best method is this:

Message Board Contracts (Free XP):
From every Message Board you find, accept all of the Contracts. The XP on them scale to your level and they are easy tasks you'll do automatically (e.g. killing a certain type of enemy or sinking certain types of ships). You can just accept them all and without having to worry about it they will complete as you play through the game = free XP. Some of the Contracts even stack so you can get duplicates from different message boards, which are always found in towns. Make it a habit to always seek out the Message Boards wherever you go and accept everything on them.

Side Quests (Best for Early-Mid Game):
When Side Quests are available in your level range, always play those. It takes 4-6 Side Quests per Level-Up (up to Level 25). After that it takes a bit longer. The good thing about Side Quests is that they are entertaining and often lead to Cultists. If you want to make things really fast you could skip the dialogues.

Conquest Battles (Best for Late Game):
When you run out of Side Quests, play Conquest Battles. There's usually 1-2 of them going on somewhere on the map, marked by a blue and a red flag. Having uncovered every sub-region on the map really pays off here. This way you can instantly see where they are happening without having to bring down a region's power. At the very least, there's always a few regions that are weakened and by killing the leader you can turn the region vulnerable and play the conquest. But obviously, you can only see the status of regions you have uncovered. Always pick the "Hard" conquest version which makes you side with the invading army. Normally, you'd bring down a region's power by burning war supplies, pillaging the nation treasure, and killing soldiers of the faction in power. But this takes too long, don't bother with that. Uncover as much of the map as possible and then look what region is already weakened. Then you don't have to go through all the motions every time and save a lot of time. The conquests give a comparable amount of XP to a long side quest.

Arena (Best for Level 45+):
By doing the story and all high-level side quests you'll end up at level 45-47. To reach max level 50, the fastest way in the endgame is to repeat Arena fights in Pephka. First you must beat all 50 champions. After that new Level 50 Enemies and random opponents spawn in the arena. You can just keep talking to the arena manager and challenge new fighters, the level 50 enemies give around 70,000XP per fight and only take 2 minutes. It's recommended to use a warrior skill build for this.

Miscellaneous (So you don't get bored farming Conquest Battles):
Another alternative is to complete all Objectives in Forts as they yield a big amount of XP, but it's only worth it with good gear if you can kill enemies very quickly. The Arena in Pephka is still a better place for fast XP if you have a high enough level.
What's special about this game is that all quests and enemies level with you (aka they scale to your own level). So even if you go back to a lower-level area it will give you quests for your current level bigger XP rewards. During the story there are also huge level gaps so the game forces you to level up before you stand a chance in story quests (e.g. Sequence 7 has a 10-levels difference). By doing the story you will probably end up at Level 36-40. After Level 40 it goes really slow. There are a few endgame areas and world bosses but once their quests are done, replaying Conquests and Level 50 Arena Bosses is your best bet to get to Level 50.


XP Boosters

The game offers time savers in form of XP Boosters. You could prepare a bunch of Contracts and Side Quests, then complete them all together after activating XP Boosters.

- A 25% Temporary Booster (2 Hours) is available for 10 Uplay Points in Ubisoft Club. It's a ONE TIME purchase only. So use it wisely. You cannot buy it a second time.

- A 50% Permanent XP Boost is available for 1000 Helix Credits (microtransaction) via the in-game Ubisoft Store.

Summary
Get tons of Message Board Contracts, do Side Quests, play Conquests when all else is done and grind out the Arena Bosses when you're level 45+.