Thanks for the thoughtful advice Everett! I went into Shopify as you suggested and discovered that not only could you hide all tags from product pages (without deleting them, which is important because this is how our products are categorized on the backend), but you could display the collections a particular t-shirt belongs to instead of the product tags.
See here for an example of a t-shirt with collections displayed instead of product tags.
If I'm understanding the power of internal links, this should help increase the pagerank for our t-shirt collections (which is where we ultimately want to drive people).
A couple of follow up questions...
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Should we now redirect tag pages to relevant collections so the PR transfers? (ie - redirect the "knicks" tag to the New York Knicks T-Shirts page?)
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Or, should we add no follow tags to all of our product tag pages?
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Should we include the collections links on our product pages? Or are we running the risk of having too many links on a page?
Thanks again for your insight and help.