First... I am not sure your concentration should be on watching your arrow flight. I almost never see mine except maybe a flash. But, when arrows are fishtailing badly they can sometimes be seen. There are many possible reasons for fishtailing. Improper arrow spine, fletch/vane contact, improper tuning, improper form including gripping/torquing your bow and/or a poor release/follow through sequence. I would suggest before you invest too much time; find an experienced archer or pro shop (though good ones are hard to find) to take a look at your set-up. With quality gear you should be able to tune it up pretty quickly. If there is an archery range near you, that may be a great place to start asking some questions.
There are a variety of ways to check your flight: Paper tuning, bare shaft tuning. broadhead tuning are but a few or you can combine a couple. Either way, the first thing to do is make sure you are set up correctly and I would start with having the bow checked out and make sure your shaft selection is correct.
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