Hi Anna, The first sentence; "today is...." is completely wrong. The second sentence is OK -ish. It would sound 'normal' to say; "The weather is nice today" or maybe "It's a lovely Spring day today". Yan is right in that we don't often say the Spring weather or the summer weather but we could say the weather in Spring is nice.

I would likely say - "The weather is nice today." or "We're having nice spring weather today"

I never heard saying that kind of sentence before. I think it does not make sense, because Spring refers to the season whether than used as adjective for the weather. You can say that the weather is nice, or emphasize you are talking about Spring, saying "I love Spring, weather is so nice today" or sth similar.