score:8
1. Banking & Money
You don't need much to open an account. Walk into a bank with your passport and your cash. Ask about opening a bank account. In half an hour you should be issued a passbook, an ATM card, and your USD converted in KRW will be in your account. Note that 500 USD, about 530,000 KRW, won't get you far.
Pick a large agency of a big bank, like the HQ of KEB Hana in Myeong-dong, or a Standard Chartered (they acquired Korea First Bank). English is not your friend in Korea. Most employees won't speak English.
But in order to reach downtown Seoul, and (see below) buy a SIM card, you'll need to exchange some cash. Exchange as little as possible – the exchange rates at the airport are not so good.
2. SIM Cards & Phones
Like for the exchange rates, there are better deals downtown than at the airport. There are SIM cards (and wifi routers) for rental, and prepaid SIM cards for sale at various counters. I usually stay a week in Seoul, so I buy a prepaid SIM card in Hong Kong before leaving, and that has me covered. But for four months, you better look into a good deal. Here are some links that should help:
A few comments. Free Wifi is truly ubiquitous in Korea. Aside from all the free networks with a passcode for clients that you will find everywhere, there are a million iptime
networks in Seoul, 99% of them open. I think it's a default setup on a common router, and the owners never change the settings. Once I forgot to get a SIM card before leaving, and I was never really off the grid. I could even connect to iptime
networks from taxis, as soon as we would stand still for a minute. Remarkable, really...
As for phones, Korea took its time but unlocked international phones finally work, at least those that are 4G. I used one of those for a while, with various phones, and it worked ok. But I really prefer the solution I have for short trips. Your Redmi Note 3 will work there.