Here is a list of skills that will support you in mining:
https://wiki.eveuniversity.org/Mining#Skills
One note- pay attention to the Fitting, Tanking, and Drone skills as those will be applicable to many other ships and occupations. Those are “core skills” and I would recommend training those first.
As for the crystals, you will need to train some reprocessing (and other) skills to use them. For example, to use a T1 Veldspar crystal, you will need:
- Veldspar Processing III
- Reprocessing IV
- Industry I
- Science II
- Mining I
Those skills will vary a bit depending on the type of crystal/ore you are looking at mining. Here is a list of reprocessing skills:
https://wiki.eveuniversity.org/Skills:Resource_Processing
You will probably not need all of those, as many kinds of ores are not in HS.
As for how far to train, a good rule of thumb is to stop at IV, unless you need V to unlock another skill/module/ship, or if you really want the extra bonus that it gives you. Use the in-game tools to see what kind of requirements different skills and modules have, and that can help you plan.