Learn Italian: Oggi sono contento.

This is a common mix-up! While you use stare for health or general well-being like in sto bene (I am well), you must use essere (to be) for emotions and feelings. Therefore, you must say sono contento (I am happy), never sto contento.

Would a female speaker say this sentence exactly the same way?

No, the ending of the adjective must match the speaker's gender. Because contento ends in -o, it is masculine. A female speaker would change the ending to an -a and say Oggi sono contenta.

Why isn't there a word for "I" in this sentence?

Italian usually drops subject pronouns like io (I) because the verb form sono already makes it clear who is speaking. You would only add io if you wanted to emphasize that you are happy compared to someone else.

I already learned the word felice for "happy". Can I use that instead?

Yes, you absolutely can! Felice and contento are synonyms. Felice can sometimes feel a bit stronger (like profound joy), while contento is often used for feeling glad, pleased, or content with a daily situation.