How should a new professor sign their name in emails to students?

Sign as you wish to be addressed by the recipient. When I sign an email with my first name, I am giving the recipient permission to address me by my first name. I almost always sign off with my first name, but in the rare instances that I don't want them to address me that way and do want to retain more formality, I use "Dr Firstname Lastname" instead. If I'm really feeling ambivalent, I'll use "Firstname Lastname", which I feel leaves the decision up to them.

I usually also have my full name with title and contact details in the sig at the bottom of my email, separated from my sign-off by a line containing only "--". This is so my contact details are handy, but also for clarity -- even if I am giving them permission to use my first name, it is sometimes useful for them to know my position and qualifications.

I would recommend signing your emails in the way that you would like your students to address you.

In my own experience in the US, most of the professors that I interacted with were quite informal with their students, preferring to be on a first-name basis in class, and likewise signed their emails in that manner (or even more informally, e.g., "-P").

Not all were, however, and those who wanted to be called Prof. Lastname tended to sign their emails thus as well. A few signed as a full "Prof. Firstname Lastname", which I found frustrating as a student, because it gave no indication as how they actually wanted to be addressed.

Note that in other institutions or other countries, the general custom may vary greatly, but I think the principle remains the same: if you sign your emails how you'd like your students to address you, it will be a clear and unambiguous signal to them, which they will likely appreciate.

Do as your colleagues.

There might be a specific tradition in your department, and you should respect it (especially if you just arrived).

If you are in a Department where the professor are referred to as "Dr. Last Name", you would seem too familiar by allowing your students to call you "First Name". On the other hand, if everyone goes by "First Name", it would seem pedantic to go by "Pr. Last Name".

And, as all the answers pointed out, you want to sign the way you want to be addressed.