the usual term used these days is "isomorphic".

an isomorphism is a bijective homomorphism. so to prove two groups are isomorphic, one way is to exhibit an isomorphism (there are other ways, but this one is probably the easiest to understand).

so, the first question you have to ask is: are the underlying sets the same size (if not, we cannot have a bijection)?

the next thing to ask is: how can we define a bijection that also is a homomorphism? well, such a homomorphism would have to take elements of order k in S3 to elements of order k in D3. an easy way to solve this "all at once" is to map generators to generators.

then check that your mapping (let's call it f) is multiplicative, that for any two elements x,y in S3, f(xy) = f(x)f(y), where the product on the RHS is the product in D3. S3 has 36 products to check, but if you have mapped generators to generators, you can get by with checking fewer (why?).