This will come pretty quickly through natural progress.
Expanding your grove awards some old coins. Once you have the ability to dig, you can find old coins that way.
The most lucrative option I found early on is once you've helped Captain Billweather (the sailor bear) a couple times. This will expand your beach and allow you to collect various shells, which can be sold to Mr Kit at the store for a good sum.
Alternatively, you can trade items to Captain Billweather and add them to your collection. These also award old coins when you select the item in your collection.
Scout Badges will also be earned with general progress and can be viewed and selected with
. These also usually award old coins.