NOTE: Before going for this achievement I would highly recommend completing the main story and getting all three contraptions, although the latter is not entirely necessary.
There are a total of 6 side missions in Wolfenstein II: The New Colossus, all of them are pretty easy and don't take very long to complete. I'm going to give a description as to how to get each side quest, as well as a description for how to complete each side quest.
SIDE QUEST #1: Destroy The Tapes
LOCATION: Manhattan Nuclear Bunker District
To get this side quest, speak to Angela Cummings who is located at Hacker Central on the ship, although I have also seen her in the shooting range. Once you get into the bunker, Go into the first room (with the two guards) and on the very far side of the room, there will be some metal boxes. Climb the boxes and in the small area directly after we climb up will be 2 out of the 3 tapes we need to destroy. Leave that small area through the door and in the next room will be the final tape to destroy.
SIDE QUEST #2: Destroy the Sumerian Tablets
LOCATION: Roswell Underground District
To get this side quest, speak to Maya Gilliam at the Hanger Bay. Once you get about halfway through the district, there will be one single train cabinet that will have its front door locked. Go around the side and you will be in the storage area of the train, two out of the three tablets you need to destroy will be laying on pallets on the ground in the storage area. the third one will be in a hallway on the second floor of the same train.
SIDE QUEST #3: Save Hercules' Cousin
LOCATION: Venus Oberkommando District
To get this side quest, speak to Hercules Leclercq (that's not a typo lol) at the 2nd floor of Set's Lab. Go through the Oberkommando district until you use a elevator. When you get out of the elevator, keep going until you see a door on your right hand side. Enter the room and interact with the cousin.
SIDE QUEST #4: Feed Rosa the Pig
LOCATION: Eva's Hammer
To unlock this side quest, interact with Rosa the pig in the pig pen. It is right next to your room on the ship. After that, go to the stairs that lead to the bar, from the diner. Take a left instead of going up the stairs and you will be in a small food storage room. Scattered on the shelves and the floor are a handful of potatoes. Pick up the potatoes then head back to Rosa and feed her.
SIDE QUEST #5: Collect Toys for Max
All you have to do is collect all of the toy collectibles for Max. There are plenty of guides here and elsewhere on where to find them. See more details on the TA page for the achievement called "Toy Collector"
SIDE QUEST #6: Get Rid of the Rat
LOCATION: Eva's Hammer
To unlock this side quest, speak to Demont Conway, who is just to the right of the Ammo Depot. Go into the ammo depot and lift the flipped over box that's on the floor.
That's it! After you've completed all 6 side quests the achievement should pop for you.