Googling for answers is a good practice: you are digging into the experience and knowledge of hundreds and thousands of people who just may have faced the exact same problem you have and can help you solve it.
As for blindly copying and pasting, well that's how we learn too. It's exactly what we do with our text books: typing out the code from the page into the IDE without quite understanding all of it just yet (if you did understand it already, you wouldn't want to see it run).
Eventually you hope that understanding will come: you're doing it right if you manage to learn something. But there is always more to learn, solving one problem just makes room for the next problem - so don't feel ashamed to get help when you need it.
If your c&p is truly blind, you will not progress; your ability as a student or professional is not measured by your capacity for Googling, but by your ability to understand and interpret the results and to develop your skills over time. Blind c&p is un-rewarding in the long run. Even for those problems where I had a magic c&p="wow, it works now!" I was never truly satisfied until I went back over it until I figured out how it works.
Sometimes you will find that Googling doesn't help. Particularly as you start to get more specialised either as a researcher or working developer, you will more often than not get an error message, type it into Google and to your astonishment see:
Your search - "Exception while unfenargling the heeble-blerg: why did you do that?" - did not match any documents.
Then will come a true test of your skills...