Before you start worrying about gear, you need to find out two things:
- What sort of rifle competition do you want to get into?
- Are there any of these competitions held at nearby local ranges?
If you can, see if you can meet up with someone via THR or another forum who is a serious competition shooter in whatever game interests you. Find out when local matches are held, and go attend one, even if you only watch. This will give you an idea as to whether or not you'd be interested in pursuing it further.
If you want to get a decent idea of the sorts of competitions available, check
this thread which is our attempt to put together a comprehensive list of the various forms of competitive shooting games, along with links to help you get started.
You can certainly get started in some of these competitions with your Mosin, and you'll be lightyears ahead of all the guys with Mosins who don't shoot them at all, or only plink, but it won't be long before you find that particular rifle lacking.