To save your current progress:

Start playing Slay the Spire.
Go to your \Steam\steamapps\common\SlayTheSpire\saves folder.
Copy the IRONCLAD.autosave (or SILENT.autosave or ???.autosave) file to a folder location of your choice.
To load your saved progress:

Quit Slay the Spire.
Go to the folder location where you copied the IRONCLAD.autosave file to.
Copy the IRONCLAD.autosave (or SILENT.autosave or ???.autosave) file to your \Steam\steamapps\common\SlayTheSpire\saves folder.
Restart Slay the Spire and click on "continue".