How do I securely fasten the connection from the pvc pipe to the strainer? I have the plastic stopper inside the strainer nut that screws into the drain tightly but water trickles from the connection. The pvc pipe can slide up and down when the nut to the strainer is securely fastened. What do I do?

I am not sure I completely understand you problem. Do you have the ferrule, a little plastic or rubber ring, between the nut and the threaded pipe? When compressed between the nut and threaded pipe, it squeezes down on the wall of the PVC to seal. If anything has a little nick in it, you will not get a good seal.

I have the plastic rubber ring with a taper on it. Currently I have the taper going away from the thredded pipe. Is the PVC pipe supposed to be loose going into the strainer?

I have the plastic rubber ring with a taper on it. Currently I have the taper going away from the thredded pipe. Is the PVC pipe supposed to be loose going into the strainer?

You have the gasket on backwards. The taper goes into the thread on the fitting.
The plastic tapered gasket should have the flat side against the compression nut . As you tighten the compression nut this compresses the gasket against the PVC pipe and seals it. It should not be loose. Good luck. Tom