Hiya Jen,
The Czech Republic is indeed part of the Schegen area so like Spain you need to apply for a Schengen visa to visit and as you have done this before then you know the procedure.
All Schengen countries require a visa to enter but once you enter initial country of your application then you can travel to any others under the validity of your visa and just to note my wife applied for her 4th schengen last month with the French Embassy and after initial 6 month validity on previous ones she has now been issued with one valid for 2 years, people have posted here that the French are the best for issuing longer visa especially if they see that you have used previous visa correctly and they don't necessarily have to be previously issued by the French Embassy.
The residence card of a family member of an EEA National is not the same as the spouse/partner leave to remain but basically is for a British Citizen who has lived with their family member ie: spouse in another european country then exercised their rights after a certain time to be able to settle in the UK.
Hope that clarifies things...