How do you say handsome in Japanese?

Saying 'handsome' in Japanese is actually quite easy and I think there are some similarities to the English word. 

ハンサム  - hansamu

If you want to say it to someone ( "you're handsome") then it would be:
あなたはハンサムですね。 - Anata wa hansamu desu ne.

Here is a video that explains how to say "you're handsome" and pronounce it.

Alternatively, you could also say:

あなたは格好いいですね。- Anata wa kakkoii desu ne.

which would be more like saying "good looking".

Hope this helps!