FileMaker 11 can convert your FileMaker 6 files into the latest version. Depending on the complexity of your original system, you might need to make some post conversion adjustments. Here's a knowledge base article to get you started:
You can try this now for zero cost, by downloading the 30 day trial version of Filemaker 11 and using it to convert your files. The conversion creates new copies of your converted files, leaving the originals unchanged so it won't affect your current set up, but you can play around with the conversion process and see how it works before actually spending the dollars to upgrade.