Getting Into the Dungeon
Getting into the dungeon actually isn't that difficult unless it's foggy, raining, dark, or you're standing in front of a bunch of ruins.
Most dungeon entrances are a small hill (about head-high) with a door in it. Clicking on the door will get you into the dungeon. Some are small mounds of stones with a dark hole in them. Clicking on the hole will get you in. Moving up really close to the hole will make your video go funny. A few others are actual buildings (like castles or towers) that have an exterior door. Clicking on it will get you in.
For the rest, you could be looking for something that looks like a cellar door or a regular door mounted on a rock or broken section of wall. It can be a little time consuming to find these entrances, but once you do, just click on them.
If it's bad weather and you have limited visibility, you can either stumble along blindly and hope to bump into it or rest until midnight and wait for the weather to change. Resting runs the chance of random encounters, but it's a lot easier than working blind.
Most of the time, the entrance is going to be more or less directly in front of you. For those few times that it isn't, look to the left and right before moving. A few dungeons (like Lysandus' Tomb) will be very close, but not right in front of you.
Before you enter, equip everything that you will need. Most of the time the area right inside the entrance is fairly safe territory, but not always. The last thing you need is to get the snot kicked out of you as soon as you enter.