(1)You can say "The weather is rainy today "  but we usually refer to the weather as IT. Examples:  It's rainy today. It's very wet at the moment. It was snowing yesterday. It's cloudy now. It's  cold and windy (2)We usually use an article (the) before the season, but it is not necessary. In the spring .... - correct In spring .... - correct I'm looking forward to summer - correct I'm looking forward to the summer - correct 'The' is more likely to be used if you are talking about a single spring/summer, but in most sentences there is no difference. (3)You don't need to say 'season' after spring/summer/autumn/winter (it is not a grammar error,  but it is a wasted word because we know spring is a season -  so we don't say it). Example: We are in spring now, I'm looking forward to summer. (4) Remember that the conjunction 'although' joins two clauses and is used to show contrast or surprise. Maybe you could say ... Although we are enjoying spring now, we are looking forward to summer even more.

Good Job!