You are treading in very murky waters. You could be asking for some serious legal problems. Before doing anything, you should consult with a business attorney and probably also with an IP attorney. there is much more to starting a business than simply hanging Out a shingle.
If they are blanks, no. If you remove a label with trademark protection. You will have problems.
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Yes, I would say being bitten would be in your future if you simply slap your own label on a product. You can certainly make arrangements with manufacturers to accomplish what I think you are trying to do. Consult with a business attorney in your area so you can have a good discussion about exactly what you are hoping to accomplish. Good luck.
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If what you purchase is already branded or protected by trade dress then, no, you may not remove any such branding and sell it either unbranded or displaying your branding or displaying any additional designs or logos on the product.
There are many apparel manufacturers who sell unbranded, generic-appearing clothes wholesale for the express purpose of having the buyer brand those clothes for retail sale. Find a few of those manufacturers and buy from them -- and ONLY them.
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A better idea would be to identify a shirt/hoodie/etc. manufacturer and have them white label your products. Not only do you want to avoid a Jansport, Hanes, etc. problem down the line, but if you intend to acquire a federal trademark registration for your logos or brand, simply emblazoning them on the front of a shirt (i.e., displaying them "ornamentally") will not work: you'll likely have to put the marks on the tag anyway.
You should speak with an IP attorney before you advance your idea.
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I agree with the comments my colleagues have posted here. Please consider contacting Lawyers for the Creative Arts, providing pro bono legal service for the arts. Their website is: http://www.law-arts.org/. Good luck!