You probably don't want to echo stuff really, and Paolo explained that quite well, but in general the best practice regarding apostrophes and quotation marks is as follows:
When you have apostrophes ' in the text, enclose it in double quotes "
echo "foo bar 'baz'";
When you have double quotes, enclose in apostrophes
echo 'foo bar "baz"';
This way you don't have to clutter it with backslashes. If you have both kinds, consider heredoc as per Paolo's example, or just stick to the style the rest of the code usually uses.
As to what comes to using apostrophes in HTML instead of double quotes.. While swapping them might be useful when you want to include variables, it would be more beneficial to always keep the same style - always apostrophes, or always double quotes.
You can also use printf (or sprintf), a function which often seems to be forgotten by PHP programmers:
printf('<input type="text" name="myname" value="%s" />', $value);