Is there a piece of software that can help me see the (hkl) planes in a crystal lattice?
Mercury (available on Windows, Linux, MacOS) has a free version and is arguably the best crystallographic viewer for beginners. It also allows to add as many $(hkl)$ planes as you want. I used the structure of calcite from COD (1010962.cif).

Open your CIF file; by default Mercury loads an asymmetric unit. Once the CIF is loaded, go to Calculate → Packing/Slicing... and tick Pack option. Alternatively, tweak the volume populated by the atoms if needed. (This also automatically populates the list of planes with the planes denoting BFDH crystal morphology.)



Navigate to Calculate → Planes... → New Plane... → hkl and add the planes by Miller indices:



and, if needed, BFDH planes one by one:



If you have CSD-Materials (paid option), then you'll be able to display BFDH theoretical crystal morphologies at once by going to CSD-Materials → Calculations → BFDH Morphology...; on export you'll get something like this:





Alternatively, there is KrystalShaper (Windows only), a free lightweight program made specifically for displaying crystal morphologies:



P. S. Thanks to Buttonwood for useful comments regarding graphical details and reminding about BFDH crystal morphology.