I cannot login to a user I just created in a PDB Just installed Oracle 12c client and server for the first time (on same machine) so please bear with me as I'm quite new to this. It's in the multitenant mode.

So I went to the server's BIN directory and ran sqlplus / as sysdba. The SID is TESTDB and looking at show pdbs; I could see that there is one PDB called BPDB1. So I switched to it with alter session set container = BPDB1 and opened it with alter database open. Now I created a user with CREATE USER testtest IDENTIFIED BY password which seemed to work.

After that, I thought I can login to the PDB from the client's BIN directory using sqlplus testtest/password. But that just gives me ORA-01017 invalid username/password; login denied.

I wonder if this is because the client doesn't know which PDB to connect to? How would I tell it this?

By the way, not sure if this is relevant but I tried setting ORACLE_HOME to the root of my client installation, but if that I got some other errors every time I try to start sqlplus. It hinted that ORACLE_HOME was set incorrectly, so I just unset that variable to continue.
You are connecting to root container by using sqlplus testtest/password where the user doesn't exist. 

Instead, you can use EZConnect or you can create a TNS name to connect to the PDB.

Example:

EZCONNECT

[oracle@host ~]$ sqlplus testtest@hostname:port/bpdb1


TNS Name

First, find out the service name of the PDB, one of the methods has been shown below.

 [oracle@host ~]$ lsnrctl status listener_name

Service "orclpdb" has 1 instance(s).
  Instance "oracdbuat", status READY, has 1 handler(s) for this service...


In my case, it's orclpdb.
Go to $ORACLE_HOME/network/admin.
Edit/Create tnsnames.ora file to add a similar entry as shown below.

ORCLPDB =
  (DESCRIPTION =
    (ADDRESS = (PROTOCOL = TCP)(HOST = hostname)(PORT = 1521))
    (CONNECT_DATA =
      (SERVER = DEDICATED)
      (SERVICE_NAME = orclpdb)
    )
  )


Now, I can connect to the PDB using the TNS name.

[oracle@host ~]$ sqlplus testtest@orclpdb