Navigating without compass

Landmarks

• Assuming you're playing the vanilla game and playthrough, you start your roaming as you exit Helgen (see image below). Directly to your right, on yonder mountain, you can see your first artificial structure: the ruins of Bleak Falls Barrow. As you continue your path according to the main quest line, you'll soon arrive in Riverwood. Bleak Falls Barrow will still be visible.

  

  Focusing on elements like these, you can start mapping the world, either mentally, or physically on paper (the latter will do a lot for role-playing!). Soon you'll realize you recognize mountains and other silhouettes of objects in the distance.

• The most prominent landmark is the Shrine of Azura, in the north east. From most locations this statue can be seen, even at night, and often provides a first clue as to where you are.

• Landmarks can be used for navigation in any sandbox RPG.

Holds

The region of Skyrim is divided into nine holds. Each of these has its own unique features, most apparent in their biotopes:

Learning to recognize these features will help immensely with navigation, especially when you know their relative positions, and can spot some landmarks to estimate your 'global' position.

Celestial bodies

Since the stars in the Skyrim night sky are a rotating static texture (Columbus, eat your heart out!), you can theoretically use these to orient yourself as well, but you'd have to learn to recognize them. The moons and sun move along different axes.
This post on Reddit has some pointers on the movements of the two moons, the sun, and the stars of Nirn as simulated in Skyrim:

Masser rises from the north-east every evening and sets to the south-east every morning. Secunda revolves a bit faster than Masser. The phases of the moons have no relation with the position of the sun like real life.

The sun revolves around the north-south axis, rising from due east every morning and sets in due west every evening.

The stars revolve around the vertical axis once every day. Not very astronomically correct, and basically the worst axis for navigation. Imagine trying to navigate with the pole star directly above your head: the stars at the cardinal directions would change depending on the time of night (though you can tell the time using Masser's position).

IRL, stars, sun, moon would all move east to west, because most of the apparent movement would be due to the rotation of the earth. Skyrim's night sky has no relations with real life.

Mods

• COMPASS

  • To completely remove your in-game compass, you can use the tweak mentioned in the section below, or use this mod.
  • To hide the compass, and entire HUD, but have it reappear when you need it, you can use iHUD.
  • To remove the markers from your in-game compass, you can use this or this mod.
  • It will be even more realistic to craft your own compass and just equip it whenever you need to know the cardinal directions.

• MAP

  • You can replace the in-game map with this paper version with improved road indication, which will add a nice layer of realism (and use flat map markers in addition, if at all).
  • Or use the maps that can be found in the world, but you might want them to be of a better resolution.
  • Better yet will be to craft a map and take it with you on your travels, and take it out whenever you are lost (if only there were maps of smaller regions..).
  • Apart from completely getting rid of markers, some locations could just be marked on your map. Custom Map Markers Redone will allow you to create up to 65 custom markers.
  • This mod combines a few of the aforementioned features: it hides the player's current position, limits zooming in on the map, and adds a craftable and equipable map (plus an appropriate mage light spell).
  • Definitely get Even Better Quest Objectives, so that the in-game descriptions of locales are a lot easier to interpret.

• MISCELLANEOUS

  • As always I highly recommend using the Unofficial Patch, which will likely save you a lot of headache, also when it comes to finding places and objects.
  • Be sure to bring wearable lanterns for your nocturnal navigation.
  • To prevent you from taking the easy way (out), you can disable Fast Travel.
  • Alternatively, to also increase the costs of paid transportation, you can use this mod. This will motivate you to go on foot or by horse.
  • Or, if you're fine with it, extend the paid transportation network by installing Carriage and Ferry Travel Overhaul.
  • For an even more hardcore experience, you can change the way walking, running, falling, and swimming work using Dynamic Realism.
  • On that note, there is a set of mods that adds a little more realism, including improved Encumbrance, titled SkyRealism.
  • To take notes along the way, you can use Journal of the Dragonbornm which is a very nice addition for any kind of gameplay.
  • Lastly, to improve gameplay during your longer travels, I highly recommend both Campfire and Frostfall.

• I suggest browsing through this S.T.E.P. guide for additional mod suggestions.

Extra

• You can turn off the in-game compass by setting bShowCompass= to 0 in your SkyrimPrefs.ini file (under [Interface]).

I see a few issues, at least for me they would be. First, not every NPC gives directions. They may saw "Those Orcs are near the village of Slobberville." The village might be 2 provinces away. I don't know about the rest of you but I would probably spend a month wandering around.

But that's just me. I follow the main quest line and try to pick up as many side quests as I can. I do however take off at times to explore. Find all sorts of things that aren't on any map. I've also wandered into some camps that I wish I hadn't.

Sounds like you don't wish to use any prompts of any kind. Just the stars. I applaud your effort. The other issue I would have is if someone asked for help to kill some big guy and tells you "I'm at the park just outside of Slobberville." For me, it could possibly be days before I answered the call.

Enjoy you journey. Take some snacks along just in case.