Thanks for your answers Fortinbras and Vert. I guess I'll be playing it this week then.
My recommendation is to play it like this:
1st playthrough: Just enjoy the game on Normal mode, don't go for any side quests, or achievements, pick whatever weapons you love to use and enjoy the beautiful story, music, and art style.
2nd playthrough: Make sure to pick Score Attack (not Vanilla New Game +). Score Attack and New Game + both count towards the "Complete New Game +" achievement but Score Attack happens to have a score overlay... that's the only difference. And there is an achievement for getting 1 million+ score in Score Attack.
With the second playthrough you can start going after side quests and other miscellaneous achievements, including the battle arenas.
It's not like Bastion on 360 though. There are a couple of hard as heck achievements involving the battle arenas. But guides online will help tremendously.
It's an incredible game. I'm so glad I'm finally back to playing on Xbox One. Past three months have been miserable for me, but I'm in a happier place now and starting to enjoy myself again. I used to not attempt to play any games after work... And I've played hours of Bastion in the past couple of days. I just got the 1000 gamerscore a couple of moments ago. This is definitely one of the best games on the system. That music!