Dwarf Fortress: Finding Lairs [Adventure mode]

Before even heading out to find a lair, there are a few things to take into consideration:

When creating a world, you can go into the detailed mode and adjust a few parameters that affect the lairs that appear.
You can fiddle with the minimum/maximum size of the lairs (bigger lairs tend to draw in more critters), the amount of lairs (if there’s many more lairs than creatures to occupy them, a lot of them will be desolate, and if more lairs than there’s room for your world will keep getting rejected), whether lairs are visible on the map ‘upon embark’ (but once revealed a lair will always be visible on the map) for which there are 2 options: ‘Only in Finder’ and ‘Always’. Select ‘Always’ because the other option doesn’t seem to do anything at all at the moment, maybe its only for reading Legends.
(Semi)megabeasts have lairs too

The very first thing you can do in Adventure Mode is to choose what kind of character you want to be. Currently you can choose between Hero and Commoner. If you choose the Hero you will get a compass in the top left corner of your HUD that tells you in what direction the closest points of interest are. If you hover over the individual points of interest, it will give you the name of the location that it is pointing to. This compass also points to lairs, specifically to the entrance and not just a vague estimation of where it is. So for an easy time finding lairs that will waste little resources, choose to be a nerdy Hero.

If you choose to be a Commoner, you won’t get the Compass. This means that you’ll generally spend a lot more time trying to find the entrance to the lair, which means that you’ll need to eat and drink a decent amount while you’re running around. If an area is freezing, it will freeze your water, but if you rightclick on an outside tile you can start a campfire (doesn’t require anything) and then you can go to the special interaction menu in the inventory and choose the option to heat up the watercontainer at the fire so that you can drink it.

Many lairs will be inhabited by nasty beings. Some of them you won’t have to worry about until you’re inside the lair’s doorway, but with others a giant roaming outside will rush at you before you’ve even started looking for an entrance. Try to sneak and if necessary crawl when approaching a site to check it out, this will reduce your visibility.

If you have enabled the revealing of lair locations, then lairs will show up on your world map. Sometimes one tile contains both one or more lairs, and another structure (like a village), and will only display a village on that world tile. But if you hover over the location, it will display any lairs that are present.
Icons of nearby and distant lairs. If you discover a previously hidden lair, it will show up on the map

When you hover over a revealed lair, you will be shown some information. The information seems to depend on what your character knows, with most lair details being public knowledge.

While travelling you will see the travel map, where you can see more specifically where lairs are located. Once you get close to a lair, you won’t be able to fast travel into it (as indicated by the red arrows in the top left corner) so you will have to switch to line of sight view to enter. This can be tricky because now you’ll have to estimate how close you are to the lair and look at your map to make sure you haven’t walked too far.
Travel map view
Annoyingly, if you open the travel map you will initially see a smaller portion of the map, and when you start moving in one direction the map will zoom out and let you see more, but it will reset when you change direction. Sometimes the smaller view of the map is an obstacle to seeing where you want to go if the lairs are too far away. For that you can temporarily wait or go to sleep which will show you a maximally zoomed out view of the travel map without having to move.
Zoomed out view while sleeping

Once you get close to the lair, you’ll have to either blindly follow the compass or start looking for entrances. I don’t know if this came in a recent update but since recently a message pops up when you get ‘close’ to a lair entrance (which can either trigger when you’re only a few tiles away or when you’re still a whole screen’s walk away).
If you accidentally skip through this message, it should still appear at the top of the HUD for a while.
Some entrances are a simple downward ramp in the middle of a field, others are doors on the side of a mound of clay.
If the lair is located in an area with a lot of uneven ground, it might be hard to spot the one ramp that leads to the lair in between all the other ramps.
Sometimes it feels like you’re just walking in circles and there’s no entrance. This might be because it’s hidden in shadow, or because from a few tiles away you won’t be able to see the icon indicating the downward ramp. In that case it will show as an empty spot.