Should is start competitive programming? Does it improve your coding and algorithm skills? Which websites would you recommend for a beginner?

Yes, i think competitive programming is one of the ways to improve your coding skills.If Participating in challenges and competitions against a lot of people is the type of learning process you like then go for it.

Some places are:
Hacker rank
Hacker earth
Code chef
Top coder
Code forces
These are known for their frequent competitions and challenges.