You don't need the article if you're talking generally about a season as a period of time. But, it's very of flexible...if you make a mistake with articles and seasons, it won't sound very strange, so I wouldn't bother worrying too much about it unless you're trying to reach a native level of speech. In your example, 'the spring is here' is fine to say too, really.
'I can't wait for it to be summer' or 'I can't wait for it to be the summer' both sound fine to me.
For referring to something specific that happened in a season, we tend to use articles, but it still won't sound too weird.
'I remember, it was in summer when the trouble came' is okay. Better to say: 'it was in the summer,' or 'it was summertime.' As well as springtime or wintertime, although I don't think we ever say 'falltime' or 'autumntime.'
'I remember it was THE summer after I graduated school that it occurred' would usually have an article as it's referring to a specific time. Even this doesn't sound too bad without the article.