How do you determine a production budget given a sales forecast for the next months

That depends on whether they just mean the amount of the product needing produced, or whether they're meaning all costs used in production.

The first is just a simple inventory adjustment. You take sales, add desired ending inventory (since you want that in addition to sales), and subtract beginning inventory (since you already have that and don't need to produce it). That's it.

If it's the second, that can get complicated and involved, and too much for someone on a forum to explain. But there is a link here that goes into all sorts of budgeting stuff:
Principles of Accounting Chapter 21