The Print function is primarily used as a debugging tool. There are generally much better ways to get output if that's your goal in a given situation. One way is to put the relevant data into a data structure--a List for example--and then output that at the end.

But. let's take your simple example and try to do what you want using a better tool: CellPrint along with TextCell (or you could use ExpressionCell if that's what is more pertinent). Try this: it CellPrints in the "Output" CellStyle since that is what the TextCell's CellStyle is set as:

TestF[t_] := 
 Module[{t0}, CellPrint[TextCell[{t, t + 1, t + 2}, "Output"]]; 
  t0 = t + 1]