National Pokédex

Filling the Pokédex is very simple but can also be somewhat complex and it does vary between methods.

Connecting to Pokémon Bank
When you transfer Pokémon from Pokémon Bank over to Pokémon HOME, it takes the National Pokédex data from Pokémon Bank and implements it into Pokémon HOME without the need to transfer every single species and form. This is very useful for various forms you may not have to hand from previous games.

Connecting to Pokémon Let's Go or Sword & Shield
When you connect to Pokémon Sword & Shield or Pokémon Let's Go, Pikachu & Eevee, it doesn't take the Pokédex data from the game but requires you to put the Pokémon into HOME. Gigantamax variants will only appear if you have the Gigantamax variant and forms that only appear in battle will appear if you have seen them.

In the Nintendo Switch version of Pokémon HOME, you have the ability to separate the Pokémon in various categories, these categories are based on the region the Pokémon was first found in

When you check the Pokédex, you will be able to see all the data for each of the Pokémon, any forms it has, all data, what games it can be found in the wild in. On mobile, you also have the ability to check the Pokémon's Ability.

On the mobile version, you have the ability to search by Moves and Abilities. This gives you a description of the move and ability as well as type and category for the moves.

With the Abilities, you are then given the option to search for Pokémon either through the Pokédex or for Pokémon in your HOME.

With moves, you can only search for Pokémon that are listed in your Pokémon HOME.

The Nintendo Switch version lets you search for Pokémon using both in the Pokémon listings but not in the Pokédex.