For starters, you won't technically be able to get rid of prices, but that doesn't mean the customer ever has to see them. You can just leave the price for everything at $0 and then edit your product type's field display settings and the shopping cart Views' settings to hide any normally displayed price element.
However, when it comes to going straight from /cart to /checkout/#/complete, it isn't as straightforward. The issue here is that the "Checkout" button on the /cart page simply updates the order status to "Checkout: Checkout" and forward the customer to the initial checkout page. There's also a "Checkout" link in the cart block that is even more simplistic, simply redirecting the customer to the initial checkout page without updating the order status.
If you remove the "Checkout" link (or if you aren't using the cart block anyways), you should be able to accomplish what you want without code. I'd add a Rule that detected if the order status was updated to "Checkout: Checkout", and if so I'd update it again to "Pending" and redirect the customer to the order's checkout completion page. One thing this won't do is trigger the checkout completion process, though, so I'm not sure it's ideal.
Ultimately, the best solution is going to be a custom module that alters the "Checkout" button on the cart page to say what you want it to say (e.g. "Confirm order now") and then execute the necessary code when clicked: update the order status, trigger the checkout completion process, and redirect to the checkout completion page.