Learn Turkish: Kışın kayak öğreneceğim. - I will learn skiing in the winter.

The root of the verb is öğren- (to learn). Because it ends in a consonant, we don't need the buffer letter 'y' for the future tense, so we just add -ecek. Next, we add the personal ending for 'I', which is -im. When the k at the end of -ecek sits between two vowels (e and i), it softens into a ğ. That is why öğren + ecek + im becomes öğreneceğim.

Why do we say kışın instead of using the normal 'in/at' ending like kışta?

For seasons, Turkish often uses a special ending -(ı)n to mean 'during' or 'in'. So kış (winter) becomes kışın (in the winter). You will also see this in yazın (in the summer) or güzün (in the autumn). While kışta would be understood, kışın is the natural and most common way to say 'in winter'.

Why is kayak in its plain form here, without a specific object ending?

Because you are talking about learning the activity of skiing in general, not a specific ski or specific pair of skis. When a direct object is general or indefinite, it doesn't take the accusative 'specific object' ending. It simply sits right next to the verb in its bare dictionary form.