Learn Russian: Отложи вещи, которые ты хочешь оставить. - Set aside the things which you want to keep.

The prefix от- conveys the idea of moving something away from a center or separating it. While it is very commonly used metaphorically for time (postponing an event), in this sentence it is used in its literal physical sense: taking physical objects (вещи) and moving them to the side.

I learned that оставить means "to leave." Why is it translated as "to keep" in this sentence?

In Russian, the concept of "keeping" a physical object is often expressed as "leaving it" (leaving it with yourself, or leaving it as it is). When you are sorting through items, to оставить them means you are not throwing them away, but leaving them in your possession. You also use the same root to tell a server to "keep the change" (оставьте себе).

Since the current lesson covers universal pronouns, could I say Отложи всякие вещи, которые...?

You could, but it significantly changes the tone. Всякий means "any sort of" or "all kinds of." Saying Отложи всякие вещи implies you are setting aside a random, mixed-up variety of things. If you wanted to emphasize that you want to keep absolutely everything, you would use все вещи (all things) or каждую вещь (every single thing), rather than всякие.

Why does которые take this specific case here?