How to auto-fill the current date in a cell when the adjacent cell in that table is filled in?

The following formula will place today's date in C3 when cell B3 is filled.

That's

IF(ISBLANK(B3),"-",TODAY())

However, if you close the spreadsheet and open it again tomorrow, it will display tomorrow's date. If you're looking to create an automatic and permanent timestamp, I'm not sure that it's possible.

It is not a built-in feature. It could be done with an AppleScript written specifically for your exact requirements and your exact table, running in the background, waiting to be triggered by data entry into any of the specific "adjacent" cells you want it to watch and filling in the specified corresponding cell with the date. You would have to start the script before entering any new data. It would not be very practical but it could be done, I think.

Probably the most practical thing would be to assign a simpler keyboard shortcut to Insert Current Date (found in the Table menu) and insert the date yourself.