Nice sentence! Here's the corrected version:
Migratory birds spend the winter in warm countries and come back in the spring.
Why this is better (in simple words):
- We add the before winter and spring because we mean those particular seasons.
- The verbs spend (for staying) and come back (for returning) are correct.
Another short way to say it: Migratory birds go to warm countries in winter and return in spring.
Try saying the corrected sentence out loud — great job!