I have a Traditional 401k that I want to rollover into my new 401k which I haven't made any contributions to yet. Can I roll my traditional 401k over to my new 401k,
Yes, but note that 401(k) accounts are more expensive than (rollover) IRAs, so really it's better to roll into an IRA than the new 401k
then make Roth contributions to my new account?
If I rolled over the traditional 401k, then made Roth contributions, would my new 401k have one section for the traditional funds and one section for the Roth funds?
Yes; you'd have two accounts (though the web site and statements might make it look like just one account) in the same plan. That's why my previous answer is "No".
To clarify: you can make Roth 401k contributions, but they would not go into your traditional 401k account; they would go into your Roth 401k account. (At the same time, you can also make traditional 401k contribs into a traditional 401k account).
EDIT: I say "two different accounts" (even in the same plan) because the pre- and post-tax money must be accounted for separately, and "accounts" are the way to do that.