'À Noël', 'à la Saint-Valentin': French holidays and their articles
In English, holidays behave simply: you say at Christmas, on Valentine's Day, at Easter, with no article. French is less tidy. The biggest holidays, Noël and Pâques, usually appear without an article: à Noël, à Pâques. But saints' days such as Saint-Valentin and Saint-Sylvestre take the feminine article la, and New Year uses the masculine le: le Nouvel An, le jour de l'An.
English speakers trip over this because nothing in English prepares them for it. It feels random until you see the logic behind each group, and then it becomes easy to remember.
When you say when something happens, French uses à plus the holiday name. What changes is whether an article comes in between. There are three groups:
Noël and Pâques behave like names. You say à Noël, à Pâques.
Saints' days are feminine because the words la fête de (the feast of) are understood. La Saint-Valentin really means la fête de saint Valentin. So you say à la Saint-Valentin, à la Saint-Sylvestre.
New Year is built on the masculine nouns an and jour: le Nouvel An, le jour de l'An. With à, le becomes au: au Nouvel An.
Rule of thumb: if the name starts with Saint-, put la in front of it. The same goes for la Toussaint (All Saints' Day) and la Saint-Jean (Midsummer).
Noël takes no article when you talk about the date. Your greeting card wish, joyeux Noël, has no article either, and the adjective is masculine.
Pâques works the same way: no article. Notice the wish, though. Joyeuses Pâques is feminine plural, unlike joyeux Noël.
Saint-Valentin and Saint-Sylvestre are saints' days, so they take la.
Nouvel An, Jour de l'An and réveillon du Nouvel An are masculine. Note nouvel, the form of nouveau used before a vowel, and du (de + le) in réveillon du Nouvel An.
The typical mistake is copying one pattern onto another: à la Noël or à Saint-Valentin. In everyday French, keep Noël and Pâques bare and give saints' days their la.
That said, Noël is not banned from ever taking an article. When an adjective describes it, you can hear un joyeux Noël or un Noël blanc (a white Christmas), and in some regions and older texts people say la Noël. The no-article rule is about ordinary date expressions.
Also, you can drop à with le jour de l'An and le soir de la Saint-Sylvestre: the article alone is enough, just as with le lundi. Finally, Halloween follows Noël (à Halloween), while the French national holiday is la fête nationale or simply le 14 juillet.
Say à Noël and à Pâques with no article, à la before any saint's day (à la Saint-Valentin, à la Saint-Sylvestre), and le or au for New Year (le jour de l'An, au Nouvel An). For wishes, remember joyeux Noël but joyeuses Pâques.