A good first step is to talk to your product liability insurer (you do have product liability??? right?
), they usually ask pertinent questions.
Unfortuatley if you want to sell worldwide then it is helpful to understand the legislation in each market - many markets have rules that follow closely with the US/EU. Less of a headache if your product doesn't transmit RF or is connected to mains. If you have competition, download their manuals/user guides and see what standards/certifications they are meeting.
If you need to test, China have some amazing labs and weirdly Japan do as well (each prefecture has at least one lab, and the testing is pretty cheap if you can get it through a 'friend'). We buy standards through the Estonian Standards chaps (
www.evs.ee), all done online and around 10x cheaper than through BSI etc..
As others have said - test as part of your design process, not just at the end!
Anyway, probably worth making sure your product is legal in your home market first.