Probably you need Richtextbox or any 3rd party control.
BTW do you want the textbox to be editable? If not(Read only) then you can try the following dirty trick(Yes with
DIV
).
<div style="border: 1px groove #000000; width: 350px;"><span style="color:red;">Red color text</span> <span style="color:green;">Green color text</span> <span style="color:blue;">Blue color text</span></div>
OUTPUT
Red color text Green color text Blue color text