I currently manage my photos on a PC using Picasa. Is there a recommended program that would let me manage them from both a PC an a Mac? Could I store the photos on an external drive connected to my router or would I use some other means?
With Photoshop the included companion application "Bridge" is used to index the files. It will display the contents of file folder from any location you select. There's the option to store the metadata for the RAW in a database of "sidecar" files in the same folder. The latter would be better if storing photos in different locations.
The Bridge UI is similar to spreading out slides on a light table to select and cull them. It also has full screen review, rating, sorting metadata functions. Not as robust in managing a large collection as Lightroom but more than adequate for my needs.
Double clicking the thumbnail in Bridge opens it in ACR. Any changes made in ACR are reflected in the Bridge thumbnail and preview when it is saved in Bridge. Changes made to one file in ACR, such as a test shot with a gray card, can be easily copy/pasted to the other files in the batch by clicking the circle icon that appears on the edited thumbnails. If you open the RAW in ACR then open from there in Photoshop any .PSD, JPG or other file you save is stored in the same folder as the RAW by default. The can be displayed separately in Bridge with a menu pick by file type.
I haven't shared the same folder with Mac and PC version of Bridge / PS for several years but I don't recall there being any compatibility issues since the RAW and sidecar file formats are identical.