The balance between weapon types in Valheim is... it's actually pretty good, overall. It's worth training with all weapons, but your own preferences and tactics are priority (so long as they work!).

You can usually make do with one weapon to use with shield, and a ranged weapon. Late game will open up further options in the latter category.

You don't need to carry all the tools with you all the time. You can leave hoe, pickaxe and cultivator at home when you don't need them. You can even use the item stands to make the swapping a bit easier, or at least I found it so.

I don't think the game actually says it clearly, but swords don't get the cleave penalty for hitting multiple targets. Other one-handed weapons (and even some two-handed) do, so that's something to keep in mind. You can check the Fandom wiki if you're unsure about weapons.