While ADO and ODBC are both Microsoft standards, ODBC is a de facto standard across most platforms. ADO is a set of COM objects (read Windows-only) that wrap ODBC functionality into tight little objects. ADO is a quick and easy way to set up database access in a windows environment. If you're programming in Windows, then you're set either way.
However it is widely considered that you'll get better performance using straight ODBC rather than using ADO.
As far as your problems connecting to your database, I don't quite understand what you're saying. You are in control of your own DSNs. If you have an Access database sitting on a server, you can create your own DSN using the ODBC Data Source control panel. You choose the ODBC driver, the server that hosts your database, and then choose the path to the database itself.