You have a little baby but you want to have a walk in both the nature and the city… In general, you need a city park. You can try the park at the Mirabell Palace.
I parked on the parking lot: Mirabellplatz 5B, 5020 Salzburg, then on foot.
In the palace, there are urban services or mayoralty, but the park is open for visiting. You can read about the palace here: The Mirabell Palace
By my impressions it is a typical city park situated in the crowded (touristic) place, where you can find quiet places
In general combine this walk with the visiting the museum in Salzburg and you will have a wonderful cocktail!
Attractions nearby:
The castle Hohenwerfen (Burg Hohenwerfen)
Gollinger Waterfall (Gollinger Wasserfälle)
The castle Hellbrunn (Schloss Hellbrunn)
The castle Salzburg (Festung Hohensalzburg)
The street of 1000 signdoards Getreidegasse
The Outlet of Salzburg, The outlet Europark