Reddit mentions of Watertown

I'm really not knowledgable in music, so I'm sure someone could give you a much better answer than I ever could, but I'll give it my best shot.

For Sinatra himself, the obvious thing would be to listen to a best of album to get started. An example would be the album titled "Sinatra: Best of the Best" or "Classic Sinatra". I don't think it's well known, but I like the album Watertown.

As far as the genre is concerned, I believe he is technically Jazz or Swing, but if you were looking for a CD at the store you would typically find him in the Easy Listening section. This is what it was at Best Buy.

And finally, you said you like Sinatra type music. This is where I will struggle to give good answers. Some different artists might be Louis Armstrong or Dean Martin. My personal favorite artist, who is inspired by Sinatra - both crooners, is Michael Bublé. If you don't know him, he's a more modern artist. Here is his cover of Sinatra's Come Dance With Me, and an Original song - Close Your Eyes. The majority of his songs are covers of older artists including Sinatra. Very talented. If you have money to blow I believe he is just starting a US tour. If you're still looking for the classic vibe, you're best bet of these three might be Dean Martin (Everybody Loves Somebody).

I hope I helped a little.

Edit: Links