The ArcGIS 10.1 Help indicates you cannot edit the attributes of Excel or MS Access files, added through an Ole DB connection, because they have no ObjectID field. You can Copy the data to a fGDB table and then edit it. Or you can script Python to edit the native MS formats.
Your best bet is to keep your geometry in shapefiles and keep the data in access. Just make sure there is a common ID in both tables. Both QGIS and Mapinfo support joining shapes to database features (eg accessed via odbc). In Mapinfo, check the function "geocode".
If you're interested in PostGIS, you might want to visit the BostonGIS website. They have great material to get you started. You can also start with OpenGeo's introduction to PostGIS workshop.
PostGIS has a shapefile importer that you can use to load your shape files. As for your accdb files, you might want to convert them to SQL first. Bullzip has this ...
According to the Microsoft support page Which Access File Format should I use?, the .accdb format doesn't support user-level security, which means there's no way to control which users can access particular tables.
A related question on StackOverflow (MS Access: securing tables from unauthorized access) also has an answer with a comment providing anecdotal ...
According to this answer, there is an ole Db provider for accdb files.
Once you have the driver installed, you should be able set up an ole DB connection in Arcmap.
You should then be able to drag and drop a table from the .accdb file into the TOC, and set up a join to it like you would any other table.
There is a Knowledge Base Article available called HowTo: Connect to Microsoft Access 2007/2010 (ACCDB) files in ArcGIS which says:
The proper way of connecting to Microsoft Access databases, either the
MDB or ACCDB formats, is through an OLEDB connection. Though it is
technically possible to connect to MDB files through the Add Data
dialog box in ...