one way:
SetDirectory[NotebookDirectory[]];
fileName = "test.txt";
file = OpenWrite[fileName, PageWidth -> Infinity];
WriteString[file,"this is a string which will show up without the quotes"];
Close[file]

To open for append and lines:
SetDirectory[NotebookDirectory[]];
fileName = "AAA.txt";
file = OpenAppend[fileName, PageWidth -> Infinity];
WriteString[file, "this is a string which will show up without the quotes\n"];
WriteString[file, "second line\n"];
Close[file]

important: must add "\n" at the end of the string. Without it,
all string will remain on same line.
You can also in the notebook, add the new line, and will be be reflected in the file. Like this:

Notice, in the above there is no explicit "\n" there, but it was added in the notebook. The above will look like this in the text file:

Export
instead ofWrite
. $\endgroup$Export["test.txt","This is a test","Text"]
$\endgroup$