This is where the years of knowledge and experience comes in. You can teach yourself After Effects, but it takes years to learn to use it to it's potential. If you've never touched the program before, you're not going to be creating anything approaching realistic snow effects without putting in a serious time investment.
Even if you do choose to go the AE route, you're going to be shooting in front of a green screen. To get a good key and to match what you're creating in AE, you need to learn great lighting skills.
It's possible to create great effects on an indie budget, you just need knowledge and experience. If you're just starting out, learn the fundamentals first. I'm not saying to not learn After Effects or whatever else you want, you just need to pace yourself. Shoot a project using a location available to you and include a small effect you can do in AE. Don't go into your very first project expecting to create full virtual sets that actually look passable; it takes time to learn.