You can polish stones using grinders and sanders if you choose to not use a rock tumbler. The benefit of a tumbler is that you can set the machine up, start it and walk away while the stones are polished for you. Other polishing methods include grinding (if you also want to change the shape of the stone) and sanding. Grinding and sanding are very similar in their methods, so we'll discuss sanding, but know that grinding follows the same basic procedures.
Put on the protective eye wear and dust mask.
Use a belt sander, a hand sander or sand your stone by hand, beginning with the grittiest sandpaper you have. Both a hand sander and a belt sander will grind the tips of your fingers off pretty quickly if you are not exceedingly careful, so if you have a small stone to work with, you may want to consider holding the stone against the sandpaper with pliers or a similar tool.
Switch the sandpaper to a less gritty sandpaper and repeat the sanding that you did in step one.
Continue switching sandpapers to less and less gritty and continue to sand with each new type of paper until your stone is nearly polished.
Finish the polishing with the very fine grit diamond-ingrained sandpaper. This final step will accomplish the fine polish you desire.