Here is a concise one-liner of code
Set<Id> ids = new Set<Id>{'0010E00000CxrgT','0010E00000CxowK','0010E00000CxugT','0010E00000CxabH'};
Set<String> idStrs = (Set<String>)JSON.deserialize(JSON.serialize(ids), Set<String>.class);
System.debug('Id Strings-' + idStrs);