I have silver cups, plates, and bowls too.
Is it safe to clean them with normal washing powder? Or Does it need some extra attention?
Also, how do I clean the layer of black tarnish/patina without damaging it?
Dish soap will not damage silverware. To remove tarnish you can get special silverware cleaning solutions like tarn-x. Note that the tarnish itself is a form if damage caused by the oxidized metal, so by removing it you are essentially scraping of a tiny layer of silver every time.
Edit - A good tip I forgot to mention: tarnish is self-limiting (meaning that only the surface of silver can get tarnished). This means that the best way to prevent tarnish is by leaving on the existing layer of tarnish. If you leave silver for 100 years and remove the tarnish the silverware will lose one layer of silver, then it will be good as new. But if silver is de-tarnished every month for 100 years, you'd have to remove 1200 layers of silver. So it's not good practice to de-tarnish too frequently; only do it when you need to.
This is what I always do and it comes out amazing http://howtohacklife101.blogspot.com/2012/02/how-to-clean-your-jewellery.html I use the one with the shampoo