When I first heard about the addition of Abyssal sites to New Eden I was a little sceptical. I instantly christened them "Abysmal Sites" and let it at that. Once they arrived in the game I largely ignore them; a mix of disinterest plus fear of the unknown (as in unknown if I'll lose an expensive ship). However, as will all things, curiosity eventually got the better of me.
Fast-forward a bunch of months. My opinion has been reformed to considering Abyssal sites as a fantastic addition to New Eden. I love the work done by the art team in them. The massive asteroids are spectacular to behold. I'm looking forward to those creeping into other areas of New Eden. The randomness of the rats in the site makes for wonderful variety. You don't go in knowing that there's x of rat A, y of rat B, and z of rat C. You have to learn which to shoot first and, once in the site, you have to pick target order quickly.
The main aspect of Abyssal sites I like most is they are "instant-on". If I have 10 minutes to spare I can jump into EVE and go run a Calm or Agitated site. If I have a bit longer then I'll jump into a Fierce. So far I've not tried the hardest of the sites - the Raging and Chaotic. It's worth noting that returning from these sites will leave you flagged as a suspect. This means other pilots can await your return and legally shoot you. If you do decide to run these sites you should do so from a safe-spot and hope nobody combat-scans the signature down.
I've you're looking for some instant gratification in EVE then I'd recommend you give the Abyssal sites a go. EVE Uni has a good article covering just about all you need to know. Have fun out there.