There are lots of websites to start competitive programming like codechef, hackerrank, hackerearth( This is good to start your competitive programming ). For GATE, we need to study OS, CN, TOC, Compilers etc. In competitive programming, we only focus on Data Structures and Algos. Here most of the things we have to learn to for competitive programming which is not included in GATE syllabus. They will help us to think like a programmer, but cannot assure you to get good rank in GATE. I did competitive programming for 3.0 yrs, it helped me to crack the toughest codes of world's best companies. So, do it for fun and learning but don't use it for GATE Prep. In competitive programming sometimes we don't need any algos or data structure but just a hit and trial method. In Gate, we need to be aware of what the question is asking and which concept it is using.
P.S: start with hackerearth, best for you