Learn Italian: Sembri stanco oggi. - You seem tired today.

In Italian, subject pronouns like tu (you) are usually left out because the verb ending already tells us who is doing the action. The -i at the end of sembri makes it clearly mean "you seem".

What is the dictionary form of sembri?

The infinitive (dictionary form) is sembrare (to seem or to look like). It's a regular -are verb, which is why the "you" form drops the -are and adds an -i (sembri).

Since we are learning stare, can I say stai stanco to mean "you are tired"?

No, and this is a common beginner trap! While you use stare for general states of health (like stai bene - you are well), you do not use it with descriptive adjectives like stanco (tired) or triste (sad). For those, you must use essere (to be), as in sei stanco (you are tired), or other verbs like sembrare (sembri stanco).

Would this sentence change if I am talking to a woman?

Yes, it would! Stanco is an adjective describing the person you are talking to, so its ending must match their gender. If you are speaking to a woman, you must change the -o to an -a: Sembri stanca oggi.