TL;DR: If you are a beginner, your teacher will probably ask you to get a practice-pad and some drum sticks.
If you are just starting as a percussionist, you will first need to learn basic rudiments, which will allow you to play a range of percussion instruments, and then specialise at some instruments if you wish to (unless you are starting to learn a very specific type of music, see below). To do this, you can just get started with a practice-pad and some drum-sticks. Although this site is specifically aimed at beginner drummers, the advice would be good for any beginner percussionist.
However, if you already know that you may be specialising in one particular area of percussion playing, you could get the necessary instrument(s). For instance, if you know that you want to become a drummer (i.e. using drum-kit), you can start with just the equipment above, but you might want to get a simple drum kit quite soon. Or, if you know that you want to specialise as a tuned percussionist, you could get a tuned percussion instrument, but I would imagine that you would still need to learn basic percussion skills first.
On the other hand, if you know that you are going to focus upon learning percussion instruments related to a specific type of music, you might instead get the associated instrument(s) quite soon. For instance, if you are studying Indian Classical music, or a specific kind of Latin, African or other world music, the basic skills that need to be developed as a beginner may require a specific instrument (for example, tabla, cajon etc…)
If you are just starting to study percussion part-time, you probably won't need much equipment to get you started; as you become more advanced and maybe more specialised you will need the necessary equipment to practice at home. Hopefully, you will be able to use your conservatory as a place to practice between lessons; this will mean you can use their equipment.
Finally, as you become more advanced, you may find that although you specialise in some areas, you will gain experience of a wide variety of percussion instruments, that you will want to have regular access to, either for practice or for performances. But even then, it is common for percussionists to borrow or hire less common instruments when needed, there are very few percussionists that will own every instrument that they are likely to play. (Although I do know percussionists with garages full of marimbas, vibraphones, drums of all sizes…) Realistically, you are most likely to gradually gather a collection of smaller percussion instruments: shakers, triangle, cowbell, tambourine, cabasa etc., and a variety of sticks, alongside your "main" instrument(s).
Note for other readers: the OP is starting a conservatory course as a beginner, having not had percussion lessons before. This is not the same as my usual understanding of a music-college, university or conservatoire course, where a certain level of previous study is expected before starting the course. (Hopefully this should save you reading through our extended comments discussion...!)