What is a good way to DM searching for something?

I am newer to DMing and was hoping to get some advice on an aspect of the game I feel poor at.

My players have had multiple instances in the story where they are looking for something / someone. Examples being:

My issue thus far, is that it feels as though its more of a chore then a cool goal to try and find the thing. They roll for perception to see if they can find stuff, and then that tends towards it.

I think an additional part of this is that one of my players has a familiar that they use to fly up and obviously gives them a large advantage at searching (though to make it very clear, I am happy they are doing that because that is a very good application and what they wanted fantasy wise to get out of their familiar).

For this upcoming search for this dragon cave, I want it to be a bit more of a game, more choices and intrigue. I've thought about getting a mountain map with a cave entrance somewhere, but I worry that walking around a map when not in combat doesn't really serve much of a purpose (because we play in person). So I am not entirely sure what to do to make searching for things more interesting and feel less like a chore.

The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.

So, if they will inevitably find the dragon cave, there is no need for an ability check; you simply say: “After searching the mountain for several days, you are standing at the opening of a giant cave. This must be it! What do you do now?”

Otherwise, it's a failure, which means the character or monster makes no progress toward the objective or makes progress combined with a setback determined by the DM.

Now, as you’ve described the challenge (find the dragon cave), there are no consequences of failure. But if you want to (“want to”, not “have to”) you can set different challenges:

But those are different challenges from just “find the cave”.

Every character likes to feel that they are the one to solve the puzzle or fix the problem. If everyone rolls perception and you hand them the answer, none of them feel the sense of accomplishment that triggers the dopamine.

I like to get as many characters involved as possible. Rather than jumping right to the Perception check I ask each player to tell me what their character is doing. A Ranger might want to look for tracks, the wizard sees through the familiars eyes, the thief might listen, the barbarian might just stomp off up the mountain. Let them all give you the actions that you can then weave together to give them the answer. Maybe the wizard does roll perception, but the ranger and thief have different skills. Use the rolls to make a story.

Perhaps the barbarian (who rolled a one) dislodges a stone higher up the hill that struck a boulder that emitted a hollow sound that the thief with his 19 heard. The ranger checks out that area and with his 17 sees some scorched stones and a drag mark that lead to the secret door. While they are searching the door, the familiar give warning that there are orcs approaching the barbarian.

I think of the dice as improv prompts, not yes/no answers.

There are many, many ways to play DND and there is no such thing as bad-wrong-fun. But with that said, DND still very much shows its war-gaming roots and it is stereotypically centered on fighting with the rules predominantly centered on fighting.

Sometimes having minor challenges along the way solved a simple roll so the party can get on with the fighting is not a bad thing. It is perfectly fine to roll perception. On success, the search is over and they can move on to the next part, which will often be a fight and to most parties will be more fun. This is quick and rewards players for optimizing and using skills their characters should have.

Obviously, if finding the thing is necessary for the plot to advance then failure cannot be permanent. But perhaps failure means they take so long that a random encounter happens in the meantime. Perhaps failure means they must camp for the night before trying again offering opportunities for attacks or other non-combat complications. Perhaps they encounter a trap or natural obstacle when they fail.

You can at least narratively enhance a search by asking the players exactly how they do it. This can help with adding complications if you want to.

In your particular case, one way they might search is the flying familiar. But that won't always be an option. If the flying familiar is a large bird, it probably can't meaningfully fly in a building, cavern, or similar enclosed area. It might even have a hard time flying in a heavily wooded area and the canopy might make flying above it unproductive.

Then they have to decide. Do they stay together to search but in a way that takes them more time which could lead to complications such as an enemy fleeing? Or do they split up, which might lead to a much smaller group being ambushed or ending up in peril? Are they in a situation that allows for a more creative search?

For that matter, even if they send out the familiar, the familiar might encounter its own obstacles.

Also, remember that the complication doesn't have to be a clear obstacle. Maybe they stumble over something and find things they weren't looking for such as valuables, a person in need of help, potential encounters they can charge into, or avoid. A complication could be an opportunity.

All of these things at least narratively add detail to the story and give you opportunities to add complications or challenges directly related to their choice if you want to.