How is it possible to trust a public key that can confirm a message's signature?
I am not commenting for what purpose your are going to use public key and answer your main question:


  How is it possible to trust a public key ...?


This one of the most important questions in cryptography: key distribution. Here are some possible solutions:


You can for instance meet Alice and exchange keys personally (but it is applicable in rare real cases).
Or you can use some channel that you trust to exchange keys.
Or you use PKI. Namely some certification authority issues a certificate confirming that the public key belongs to a particular person. The advantage is, that you can exchange certificates also via insecure channel. But to use PKI you need of course trust particular certification authority.