All of the following types of areas are the best to target when going for the Delver achievement. These areas do not require you to go wandering around an in-ground or underground maze. Most of them only require you to go in and kill a small number of moderate or weak level enemies. The only exceptions are the Dragon Lairs or Nests (not sure what to call them), as these seem to require you learning the shout the dragons are guarding. My best advice for finding these would be to buy a horse and ride around. You will see the icons appear on your compass. If it's an easy-to-clear location like I mention below, hop off and clear. Otherwise, ignore it or simply ride up to it so that you can fast travel there in the future (not a bad idea as it may have a mission or side-quest associated with it in the future).
Note: While I am not aware of any of these areas REQUIRING yo to sign up for a quest in order to clear it, it is possible that a small few are side-quest related and may require you to complete the part of the quest in the area in order to clear it (like go and get *insert random name*'s sword back).
A quick and easy way to clear a few dungeons is to target the areas that pop up on the map and your compass as pine trees. These areas typically can be cleared easily by killing a few bears or whatever other hostile creature inhabits the area.
The next easy targets, depending on your level and equipment, are the giant camps, which appear on the map and compass as mammoth skulls. Just run in, kill the giant/giants, and you're done. I kill the mammoths after the giants for the tusks, but as far as I've seen, that's not necessary.
Another target that's quick, but not necessarily easy (again, ease is dependent on character level and equipment), is the dragon lairs or nests. On the map and compass, they look like dragon heads. These are cleared once you kill the dragons and learn the shouts that they are guarding. You do not necessarily have to kill all enemies in the lair/nest. I found a high-level enemy in one (Shearpoint I believe) that kicked my arse. I decided I wanted my shout, screw killing him. Ran past him as he started getting out of his coffin, learned my shout, and sprinted away. This cleared the area for me (don't worry, got my revenge a few levels later when I was stronger and better equiped). A lot of these, if not all of them, can be assigned as a bounty quest from the Jarls in the various cities, which makes finding them easier.
Speaking of bounties, the Jarls will also mention bandit camps nearby that are typically easy to clear in a short amount of time. These camps look like tents on the map and compass. You can clear some or all of these without the assigned bounties, but that's extra gold for free from the Jarls. Why miss the easy money? Not sure if you have to kill all bandits in the hideout or just the Chiefs, but if you're there and they're attacking you, why not kill 'em all to be on the safe side?
I've also noticed that certain ruins (the ones that appear as on the compass and map looking like M's) are pretty easy to clear as well. They are to be handled the same way the Pine Tree icon areas are. Go into the area and kill whatever monster is inhabiting it. Easy and fast cleared dungeon.
Yet another easy map Icon to target are the Tower shaped Icons with a triangle flag on top. These are not like the ones without the triangle flag, as the ones without the triangle flag have inside areas that may need to be cleared. Target the ones with the flags on the icon and it's as simple as the Pine Trees and the M-Shaped Ruins.
Finally, just play the game. Side-quests and main story missions constantly send you into the larger underground dungeons. They are time consuming, but fun, so if you follow my advice and find yourself close to the achievement (check your progress by hitting start, right trigger to go to general stats, and looking at "Dungeons Cleared"), just hit up a few side-quests and you'll have it before long. Enjoy the game, I know I am!