If the breaker has a yellow test button (right side of panel), replace it with one (same amperage as what you remove) with one that looks similar to what you have on the left side.. no test button. Looks like Eaton Cuttler Hammer brand.
Then, at the outlet, replace it with a GFCI like below (this is a 15A version, it will work if your wiring and breaker are 20A...however the opposite is not true, don't put a 20A GFCI receptacle on a 15A circuit). If you have two outlets, you have some different options.
1) You can get one GFCI outlet and attach power in to LINE (always power in) and the next outlet to LOAD. This will protect anything on the LOAD side with GFCI. HOWEVER, if it trips, all outlets on the LOAD side will turn off. Similar to how your breaker setup is.
2) You can get 2 GFCI outlets, and when running the wire only connect to the LINE side, and then connect the next outlet from the LINE side on one to the LINE side on the other. This gives each individual outlet GFCI protection, and if one GFCI trips, the rest of the outlet will still continue to run.
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