New clutch installed and while driving there are significant vibrations when shifting?

No, this is not normal for a new clutch, it sounds like they have the wrong flywheel on the engine which will cause the vibration. Also, it sounds like they did not plug in the VSS sensor, which is why the speedometer is not working. Do me a favor and start the engine and with the transmission in neutral rev the engine is see if the vibration is there. Please upload a short video so I can see so we can be on the same page. Here is the location of the VSS sensor so you can see if it is connected.

the VSS is located on the right side of the transmission.

Let us know how it goes.

So the noise I’m concerned about happens mostly between 1st and 2nd gear. It’s a clunking noise. I am thinking the guy who did this may have not tightened the bolts to spec and the pressure plate maybe uneven or maybe the release bearing needs adjustment? Not sure how to check for that. I do notice a little fluid coming from the slave cylinder but not enough to where the master cylinder has ever run dry so I don’t think there is air in the lines. The clutch fluid looks good, I don’t think the noise is coming from the master cylinder. I did find the connector on the transmission and it was unplugged so I plugged it in but the speedometer still isn’t working.

Oops, my bad the speedometer is working now.

The starter noise is because it sounds loose (mounting bolts) or it needs to be replaced, I would check the starter mounting bolts. Also, the noise you hear between first and second sounds like the sub frame or engine mounts are loose, I would not drive the car anymore and take it back to the place that did the job and have them recheck it. The vibration is probably from the loose components. Thanks for the video, it helps.