Many pastures, ornamental lawns, parks, golf courses and homes have soil that is considered wet. With cedar fencing, the most common, there are almost always water considerations when building fences. To properly set wood posts in wet ground, you need to use concrete and pressure-treated wood. There must be enough depth to penetrate below the frost line, and adequate backfill for drainage. Done right, posts will last for years in any kind of soil.
Dig a hole 18 inches deep and 6 inches wide with a post hole digger.
Shovel gravel into the hole to a depth of 6 inches.
Pour a small amount of cement into the wheelbarrow. Add water and stir with the shovel to make the concrete workable. Shovel the wet concrete into the hole to a depth of 6 inches.
Plunge the post into the hole forcefully. Twist and push on the post to penetrate into the wet concrete until the post bottoms out.
Level the post by holding the level vertically against the post from both sides. Use the shovel to add more wet dirt to the hole as you continue to check your vertical alignment with the level. Finish by filling the hole to the top with wet soil. Tap down the soil with the end of the shovel. Check the vertical alignment again and straighten if needed. Wait 24 hours for the concrete to harden.
If you live in the northern part of the US, the frost level could be reach down to 24 inches, and you would need to make your hole that deep. Check with local authorities for your frost level.