This is one of the most asked question in the chat. This guide is to truncate the amount of questions and to help the people in need.
So first of all, when you get in the quest an NPC will ask you to register on the computer beside your bed which is on the right. Your roommate is your rival. Who we all know is very annoying and mean. Anyhow, when you interact with the bed your Pokemon's will be healed. This will help you while you fight the NPC's, check out the rooms in the floor and be sure to check the dustbins!!
Get in all the rooms in the Floor 1 and be sure to interact with all NPC's they give you hints on how to complete the quest. Also as I said before, be sure to check the dustbins! Beat all the NPC's you can get, this will help you beat the third gym without having to go into the wild and train, this is also a good way to get some cash in order to save up for a bike if you want to get one. You can always go back to your room which is below the first floor.
So before talking to the Doctor in the kitchen, talk to all the NPC's and check the trash, after you check the trash bin's you will stumble upon a Pecha Berry (If I recall correctly). Then go to the Doctor who is in the kitchen which is located on the left side before the exit. You can ask him how you can help by using the Pecha Berry, and he will ask you to go to the second floor and meet the actual Doctor. Before going to the Second Floor, a good thing would be to go heal your Pokemon's from your room so you will be fully prepared to battle the NPC's in the Floor 2.
So after you get on the second floor, head down to the rooms in the Floor 2 and battle the NPC's, the last room in the Second Floor is where the doctor is located, talk to the doctor and he will ask you to go meet the Blissey in the ballroom, which is located in the Third Floor. After exiting out of the Doctor's Room, if you go right and upwards, you will find the guard and he will say you cannot enter yet. So head down and you will find a trashbin, interact with it and it will give you a leftover. Again, go to your room and heal yourself.
Floor 3 ( First room)
There are two rooms in the Floor 3, one is where your rival is located in. He will battle you so be sure to teach him a lesson on not to being a bully! After you beat him, he'll go from the SS Anne (If I recall correctly)
Floor 3 ( Second room)
This is the ballroom the Doctor was talking about, if you want you can talk to the NPC's there. After entering, go left and head upwards from the stairs and take a left turn after you reach the top of the stairs. After taking a left turn, head downwards and you will find the Blissey the doctor was talking about. Get the potion for the SS Anne's captain from her and head to the second floor.
Floor 2 (Captain's Room)
After reaching the Floor 2, go to the guard whom didn't grant you access to the captain before, he will let you enter in. Interact with the Captain and he will give you the TM Cut, which will help you get into the Third Gym.
If you have any questions or if something is wrong, please let me know!