Hey Eric,
You absolutely can. You will need to go in to the admin section, go to Tags, then click the New dropdown and select New Tag Type. I have not heard of a maximum number of tag types, so I magine you can create quite a few. From your example, it sounds like you would need to create at least three: Products, Speakers, and Topics. You can also add new tag values to existing Tags. You should be able to use all your different Tags within a Program, just be cognizant that you will only be able to select one value for each tag.