What is your location? if you are in the US the 1060 6GB and the RX580 8GB are pretty much the same price at like newegg (depending on brand that is).. they do have a sale on now though that may be of interest if you are in the US that is..
for example the RX580 8GB is
https://www.newegg.com/Product/Product.aspx?Item=N82E16814131712 only about $230 (pre tax if there is any with8 the discount that ends in a few days)
on newegg the 1060 6GB is right around the same price the RX570 8GB card from powercooler is $150 (on sale ends in 3 days)
Of course this is all if you are in the US.. if not prices wil be different for sure. IF you go for a AMD card ( I have the RX 580 8GB card) the difference between the 570 and 580 (8GB editions) is not too much.. though if you can afford it get the faster card if you plan to use it for a while. BUT the 570 is definitely not slow in comparison. if you are on a smaller budget I would just get the RX 570 as its not a lot slower then the other 2 cards (if you are getting the 8GB version.. the 4GB version is cheaper in some instances but in some games the lack of Vram will cause issues / slower performance.
The 570 is close to the performance of the 1060 6gb and the 580 8GB is pretty much the same or a little more in some games a little less in others..
SO in short get the RX 570 if the difference is a lot (and its the 8GB card not the 4GB card) if the 570 is a 4GB card and the 580 is a 8GB card then there will be more differences.
any of those cards will be much faster then your current 760 card
Also a reminder.. if you have a SLOW cpu being you have an older GPU.. it may effect the performance a bit.. (if you can afford it get the fastest one you can afford) unless you don't plan to keep it for a long time. It is always what I do.. get the best I can afford because I want to keep it for 5 to 7 years before I need to think about replacing it with a faster GPU.. so like having to spend an extra 100 now to get another couple years of use from the card for me its worth it.. BUT like I said not a lot of difference between the 570 and 580 (if you are referring to the 8GB cards)