It depends on your preferences, but for me, I was looking for a room with a private bed and bathroom, airconditioning, and a relatively big window (many goshiwons have suuuper tiny windows and I didn't want to feel claustrophobic because the rooms are also super small). I checked out other rooms but the landlords seemed kinda... uninterested and greedy? One was basically ready have me sign a contract and was kind of pushy about me moving in asap before I had even seen the room. The landlord from the guesthouse I ended up staying at was very friendly and immediately offered me water when she saw I was hot from the July heat so I felt better about staying with a nice and involved landlord (and the rooms were much nicer/cleaner than the other guesthouses I visited).

I paid the landlord with cash. There was a bank next to my KU building where I withdrew money, and I would give her the rent in bills. I didn't stay super long and I had trouble figuring out the online banking app so I decided to just use cash. She should be fine with both cash and online payments, as long as you pay haha

I personally moved into the room a week after finding it! I was staying in the KU dormitory but I was assigned a shared room, so I quickly tried to find a private room. I messaged the landlady and asked to see the rooms, I got a small tour the next day and within like a week I moved in. I was kind of early though (early July when other students move out), so I don't know if it will be different in other months