Welcome to Eldemore, Pekopekoyama! It's nice to meet you, and I hope you enjoy yourself here!
It looks like you've already adopted your October monthly pets, so you've got the "Adopt" tab covered unless any special releases are announced. The "News" tab will glow white if there are any unread updates, so make sure to check that once in a while and you'll be set to go. You can view the pets that you already own from your "Homestead" tab, as well as create groups to organize them if you feel like it. The "Codex" tab serves as an archive and collection of bits of lore about the world of Eldemore. You can create a wishlist of pets you'd like to trade for there! The Trading Post is most useful to see when you have a new trade, when it will show a glowing number to indicate how many unseen trades or trade notifications are waiting for you. The "Forum" tab and "Eldsonas" tabs are fairly stand-alone, so you don't need to use any of the other tabs to use them fully. The "Bazaar" is the site's premium-currency store, where you can spend baubles if you have them.
If you have any questions, please feel free to ask! I hope to see you around!