To set up a bank account, you need to have a visa that is valid for longer than 90 days, meaning you cannot open a bank account if you are on a tourist visa. Some bank branches might even reject you if you are on a visa that is six months or less, but this seems to depend on the bank and branch in question. If you get rejected, maybe try another branch or bank and you might get lucky. Another requirement for setting up a bank account, that seems to have come about recently is, that you now need to have stayed in Japan for longer than six months at the time of application, or you need to have already found work and submit a copy of your contract when applying. This is a recent change in accordance to a new anti-money laundering law and detailed information is still somewhat scarce at the moment, but it could mean that getting a bank account when you first come to Japan has become a lot more troublesome.
What you need to bring when you want to set up an account will be slightly different, depending on the bank, but in general, you will be required to bring your resident card (sometimes passport might get asked for too), a certificate of residence (juuminhyo, 住民票) and your personal seal/inkan/hanko (though some banks might allow you to simply sign). In many cases, you will also be required to provide a valid phone number when opening the account, which can be a problem. If you do not have a personal phone number yet, use a friend’s phone number, or simply give your companies or schools phone number for the time being. You can always change the number later.
Once you filled out the necessary forms at the bank and submitted them to the person at the counter, you will receive your bank book (tsuucho, 通帳) and a cash card (kyasshu kaado, キャッシュカード) sometimes on the spot or in rarer cases via post, roughly one week after opening the account. Some banks might not issue a cash card automatically and you will need to specify that you want one when making the application. Also note, that just because you have an account at a bank does not mean it will be easy to a credit card. Getting a credit card as a foreigner in Japan can be pretty difficult but at the same time this is seemingly entirely up to chance though, some people appear to be able to get a credit card soon after coming to Japan, other people that have lived in Japan for many years will still complete most of their transactions in cash because they do not have access to a credit card.