If your terminals are covered in a blue-white or blue-green substance, then they are definitely corroded -- this happens when batteries are left in the device for a long period of time.
There are six screws you need to remove in order to get the case apart. If you're lucky, these are normal Phillips screws -- but Nintendo used Y-shaped security screws on some Game Boys, which require a different bit (I've linked to it at the top of this guide).
After removing all six screws, carefully slide and disconnect the ribbon cable (pictured) that's connecting the two halves.
There are two screws holding the logic board in and two screws holding the audio port on. Remove all 4 screws and carefully remove both boards (and the power button!) from the back case.
We're going to remove the three battery contacts/terminals using a small flathead screwdriver so that we can clean them more easily. There are two removal contacts at the bottom of the battery compartment and a single centered one at the top of the battery compartment.
To remove these contacts, use a small flathead screwdriver to get behind the contact and then extract it using a pair of needle-nose pliers. This process is made super easy because the clips are easy to access from the back. Carefully remove the top contact and two bottom contacts.
To remove the corrosion, soak the three contacts in a small container of distilled white vinegar. You may need to scrub them slightly using an old toothbrush or your flathead screwdriver to get them squeaky clean.
Let them soak for about five minutes, rinse them with water, and then soak them in rubbing alcohol for about a minute. This will remove any impurities created by the vinegar. Finally, dry them with a paper towel.
We removed the back of our Game Boy so that we can clean the two non-removable contacts at the top of the battery compartment. Dip a bit of paper towel in some rubbing alcohol and carefully clean the two contacts that are soldered to the board.
Make sure everything is dry. Replace the contacts, put the case back together, insert the batteries, and test your Game Boy out!