For general materials, like iron and diamond, digging while being patient is your best bet. Alternatively, cave exploring might be more your thing. Note that certain materials only spawn below certain Z levels, so if you're not finding what you need, you may need to go deeper. Diamond and redstone, for example, are easiest to find on the same level as lava.

Once you have a diamond pickaxe you can mine obsidian (pour a bucket of water on lava) which is needed to build a nether portal.

Materials:

Coal and emeralds are usually higher up in caves, where stone is (emeralds much less apparent, best obtained through trading with villagers), and are rarer the further down you go. Iron, copper, and lapis lazuli are found almost throughout all height levels of cave. Gold, diamond, and redstone usually start around where stone-meets-deepslate, and further down. Ancient Debris is only in the Nether, and far down in the layers of netherrack.

Some non-ore materials, like bamboo, dripstone, moss, amethyst, coral, prismarine, etc., just require exploring the right biome to find them.

"Templates" are usually in chests of points-of-interests (below).

Points of Interest:

Really the best way to find most of these are just exploring. If your world is oceanic, consider a boat or boat-with-chest. If the world is continental, maybe consider a horse/donkey/mule/camel (if you stumble across a saddle).

Some PoI's are common-ish enough where you'll probably find at least one just by exploring, or exporing the right biome. Shipwrecks and underwater ruins in the oceans, pyramids in the desert, villages and incomplete nether portals in most biomes. These you just have to stumble across them, and you'll know when you see them.

Some PoI's are very hidden, and really require luck and time to find, like "old-style" dungeon spawners, Ancient Cities, Trail Ruins. Trail Ruins in particular, you can spot them from the surface if you notice the terracotta and/or suspicious gravel (which looks like gravel at a glance, but it may be out-of-place next to dirt/grass), but they're pretty rare to find. Ancient Cities are always in the Deep Dark, but requires a lot of digging/wandering around blindly to find due to the rarity. The old dungeon spawners are very obvious if you stumble across them, it just looks like a cobblestone cube with mossy-cobblestone mixed into the floor, and the spawner cage in the middle. Nether fortresses look like bridges and buildings of dark-red netherbricks. Piglin Bastions are giant towers of blackstone (plain, bricks, etc), with some guilded blackstone scattered in.

Strongholds, can be found with Eyes of Ender (obtained later in the game, after killing Blazes and Enderman for their Blaze Rods and Ender Pearls respectively).

Some, like Woodland Mansions, Ocean Monuments, and Trial Chambers (in 1.21) can be found by leveling up a Cartographer villager and purchasing the corresponding map from them.

There's a block called a cartography table, place that in a village and trade with the villager that uses it. Eventually they will trade you special maps that tell you the location of certain structures. That's a good way to find some side structures.

Caves are great, big snow capped mountain caves are better when you go deep ;)

Just going to add that you can also use commands or external tools to locate structures. Some people would consider it cheating, but imo, as someone who has played the game for a long time, I think there are times where using such tools can make the game more fun (which is the goal of a game). It can be fun to just explore around, but it can be frustrating not finding what you are looking for.

If you are an operator on the server, you can use /seed, which will give you the world seed that you can copy by clicking the text in chat. Put that seed, when prompted, in to https://www.chunkbase.com/apps/ and it will tell you the coordinates of the structure you wish to find.

You can also use the /locate command, info for using it is here: https://minecraft.wiki/w/Commands/locate

Of course, I’m only mentioning these as options. It’s up to you what you think would be most fun for you and your kid. Just walking out in to the wilderness can also be fun (remember, your spawn is usually within 100 block range of coords 0 64 0, so you can use that to get home). Others have suggested cartographers; if you need Emeralds selling a Farmer his own wheat (from village hay-bales) can help.

You can make a stair-like pattern and dig down a bit for rarer resources.

Other than that you can check where certain things spawn in the Minecraft Wiki. Some things only spawn or are more common in certain biomes.

I also run in a straight line when I first start a new world until I find some resources that I want. I don't have any gear at that point so it's pretty low stakes.

If you really want to get into it you can enter your seed into various services and it will tell you where things like villages are.

Explore. Finding minerals, the best way is to go cave diving, find a big ravine and set up camp nearby.

For finding structures, explore more. Assuming you have a big waterway, make a boat, and start paddling. Leave yourself breadcrumbs to find where you started.

Depending on the age of your kid, you might want to turn on coordinates, so you can find each other when you get lost.