Forget the transient output signal at the moment.
Try to bias and size your transistors in a way that all are in saturation and you reach your specs (mainly the gain,transit frequency,phase margin,ICMR and output swing).
You can refer to some tutorials from Google (like **broken link removed**) or read some basic analog books (Gray-Meyer,Razavi,Allen-Holberg etc...) and see
how things work.Don't waste a lot of time in bibliographic research,just get a good perception of how it's parameter affects your performance and then play with the simulator to achieve your goals.
In this way you have a good theoritical background and an intuitive perception of your design as an IC engineer.
The final transient test should contain a sinus input source with amplitude=max. input signal (given from your specs) and if you notice clipping or distorion then you should revise your linearity and output swing goals.
As i see now,transistors MN4,MN5 in your schematic are not biased...use an ideal Vdc there or a current mirroring.