The first is right You can also say "Today 's weather is sunny", "Today it's sunny", "Today is sunny", "Today's sunny". You would say that to someone who could not see it (a phone call, email etc). If you're talking about the weather during the day that is still to come, you'll probably use "will be".
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.