First, please note that the save state feature in WinUAE may not be reliable when using hard disk emulation - see note at the bottom of this page.
Load up your game as normal.
When you want to "save" press F12.
Select Misc from under Hosts (arrow a on the screenshot to the right).
Click Save State… (arrow b).
Next type in a file name (arrow c)
Click Save (arrow d)
When you want to re-load/restore a save state file then:
Press F12.
Select Misc from under Hosts (red arrow a).
Click Load State… (blue arrow e) and double-click on the file.
If using hard disk emulation, try to ensure you:
avoid using HDF files (use hard drive directories instead)
avoid using RTG (Picasso96) emulation
avoid emulating 'expansions' (e.g. uaescsi, bsdsocket)
If you don't understand the previous three bullet points, maybe it would be best to stick with floppy emulation for save states.