Totally Reliable Delivery Service

Before proceeding, note that you'll either have to purchase the Stunt Sets and Dress Code DLC, or you'll have to join a lobby in which the host has the DLC; otherwise, you'll be unable to equip several of the power-ups.
As of 4/23/23, equipping a host's power-ups does not pop the achievement if you don't have the DLC yourself (even if you do their corresponding events, despite what one commenter suggested). The devs also removed the world pick-ups for any of the power-ups, so you'll potentially have to do all of the events so they're accessible at the power-up station.

EDIT 4: They did yet another update that broke this achievement for those that haven't purchased the DLC. As of now, it seems like you'll either have to purchase it or wait for another update that potentially fixes it.
EDIT 3: According to FFX Brotherhood, they updated the game, and people that were previously unable to complete the achievement are actually getting it to pop. I can further confirm this since I helped one person get it that was unsuccessful when it first came out.
EDIT 2: If you don't have the DLC and the achievement isn't unlocking for you, try completing each of the events on a local save prior to joining any public lobbies. Equipping all of the power-ups after that worked for other people that were having problems.
EDIT: Several of the comments have stated that they didn't get the achievement even if they joined someone else's lobby and equipped them. My guess is that you would likely have to go through the process of completing all of the events with the host as a way for the game to recognize that you earned them, or the game really just wants you to buy that DLC. ¯\_(ツ)_/¯

So to start things off, most of the power-ups are unlocked by participating in events, which are marked on the map with a green square with a flag. There are technically only two events that you'll have to complete. They're both relatively easy, but there's also an optional third event that you can do just for the sake of having one of the power-ups unlocked, but it's not necessary for the achievement.

Power-ups can be equipped at the purple/pink lightning bolt on the map after you purchase the stores from the wizard in an area. They are as follows:
• K.O. Helmet - Can either be found on Stunt Island in front of the house surrounded by castle walls, or it can also be unlocked by completing the Laser Barrage event in Down Town (the starting town).
• Green Floaty Thing - Can be unlocked by completing the Teddy Seek event in Cheek's Crest. Dress Code DLC is required.
• Wings - Can be unlocked by completing the Windy Wings event at Windy Butte (top of the snowy mountain).
• Boxing Gloves - Can be equipped from the power-up station assuming you have the Stunt Sets DLC. Also can be found in the red studio building on Stunt Island.
• Boots - Can be equipped from the power-up station assuming you have the Stunt Sets DLC. Also can be found on top of the volcano on Stunt Island.

Once you have them all unlocked, you'll just have to go to one of the many power-up stations and equip all five at once, and voila; the achievement is yours.

If you want to see how to do or even find the events, here's an optional video tutorial (credit to Trophy Boiii on YouTube):

You need to equip 5 power ups. You start out with 2 so you need to complete events to unlock 3 more

I show and explain it all in the video :)

Too Powerful! - Achievement Guide:

This is a very niche solution, but it may apply to some of you out there.

If you're friends with someone in real life that owns the DLC required to obtain this achievement, and you AREN'T game sharing with them, go to their house and play the game.

At the Xbox dashboard, your friend should sign in first, then you sign in and YOU launch the game. You might have to play the 3 events to unlock the helmet, wings, and inner tube floaty. Successfully complete each event, then go to a power-up station and equip all 5 to unlock the achievement.

FYI this solution still works if your friend has their Home console set to a different Xbox than the one you're playing on.