Tumbled stones are usually shinier and look polished. Points are available (like my citrine with a honey gold tip), raw aka "natural" stone, clusters, and geodes.
I would recommend quartz, rose quartz, amethyst, citrine for starters. I have an iolite piece that I am currently very fond of and was happy to add to my starter collection. I will be adding labradorite, moonstone (the non rainbowy variety), and flourite to the group as soon as nice samples become available in the store I shop at.
Don't be afraid to "test" the stones you want to buy, and don't hesitate to put down those that don't ressonate with you. I found a strange piece of amethyst that tingled with an almost electric spark. My iolite does the same but to a lesser degree