This is not possible to do with a textarea. You are looking for a content editable div, which is very easily done:
<div contenteditable="true"></div>
jsFiddle
div.editable {
width: 300px;
height: 200px;
border: 1px solid #ccc;
padding: 5px;
}
strong {
font-weight: bold;
}<div contenteditable="true">This is the first line.<br>
See, how the text fits here, also if<br>there is a <strong>linebreak</strong> at the end?
<br>It works nicely.
<br>
<br><span style="color: lightgreen">Great</span>.
</div>