How to Get a Wooden Bucket and Start Using Bucket Lib
Bucket Lib itself adds no items, so the first thing you actually want is a bucket from a mod that depends on it. The most common one is Wooden Buckets, which is loaded on demand by Bucket Lib.
Confirm the dependency is present
Open your mod list and check that Bucket Lib is installed alongside the bucket mod you want to use. If the bucket mod is missing, nothing will appear in your crafting grid, because Bucket Lib only provides the shared code that other mods hook into.
Craft and test a bucket
Once both mods are loaded, craft the bucket using the recipe shown in JEI or REI for your specific bucket mod. After crafting, right-click a water source to fill it and right-click again on a solid block to empty it. If the bucket fills and empties normally, the library is working.
Check that other fluids behave
Try the same bucket on lava, on a modded fluid tank, or on a cow. Bucket Lib handles fluid registration and milking logic behind the scenes, so any bucket built on top of it should accept water, lava, milk, and fluids added by other mods without extra setup. If a specific fluid refuses to go in, that is a configuration choice made by the bucket mod, not a bug in the library.