First, forget about Access. SQL Server is a server, like a mail server (Exchange) and so there isn't "a user" or a default.

It doesn't matter if SQL is on the same pc or another, you connect the same. With Express you need to be sure that remote connections are enabled, but the connection is the same. Use TCP, 1433 should be the default, but if you use Express and might end up with multiple instances, be sure that you pick a port that's unused and document that.

The local administrator on the SQL Server machine is an administrator in SQL Server by default. That's called a sysadmin in SQL Server. Connect with that account and then you can add a "login" that you can use to connect. If you want to expect some Windows group, like Power Users, and use NT authentication, then you can connect with SQL Server specifying a trusted connection. If you want to use a name/password, you must create a login (with password), and then create a user in the database where you will store your data.