It won't matter that your debit card has a Visa logo on it. Hotels and rental cars will still recognize it as a debit card. It happens to run through the credit card processing networks, but it's still a card that anyone (even with a bad or no credit history) can obtain, and people (especially those who don't have credit cards) tend to have less in their checking accounts than most credit cards have in their available credit line.

Just to give you a flavor of the restrictions you may face, in Anchorage, rental car deposits on debit cards vary from $200 to $500 depending on the company (one doesn't accept debit cards at all, and most agencies require a copy of a flight itinerary [one also requires an insurance card, and one runs a check of your credit history in lieu of this extra paperwork])--this helps to screen the less risky travelers from the more risky local residents. Credit card holds are significantly less than debit card deposits and don't require any additional qualifying paperwork. Most of these agencies here are franchises, though, and their policies are different from most corporate-owned locations (whose deposits aren't quite as high).

I'm not familiar with hotel debit card deposit practices, however. Some hotels may accept your debit card and treat it as no different from a regular credit card. Others might not accept them at all. Most will probably be in the middle--they'll take it, but they'll either charge an additional amount, or at the least, instead of doing it as a temporary authorization (which places a hold on your funds), they'll actually charge the amount out of your account and will then refund it upon check-out.

Of the rental car agencies in Anchorage that accept Priceline/Hotwire prepaid deals (I'm only aware of two), if you don't fulfill their requirements, they will deny you the rental. I don't know if you can get your money back from Priceline (and if so, I'd imagine it's quite an ordeal).

Your experience will probably differ significantly from what I've posted. I'm just giving you an example of what you might face when trying to travel without a credit card. If you plan on being in the U.S. awhile, it might not hurt to try to establish some credit history here and get a real credit card--check into student cards (United offers a College Plus Visa, IIRC) or a secured credit card through a local bank or credit union.