Is this paragraph grammatically correct ?

Currently I'm taking IELTS training and planning to move to Canada by May this year on a student visa.
One answer suggests your sentence is a run-on. Please don't worry: it isn't. It would certainly be possible to compress it, but there is no particular need to do so, as the basic skeleton is correct.

No, I would rephrase it. I would say something like, "I plan on taking the IELTS test so that I can obtain a student visa and start studying abroad in Canada by May of this year."

No, it is a run-on sentence. 
I would write it like this:  I am currently taking IELTS training.  My plan is to move to Canada, this May, on a student visa. 

Mostly correct, but I think this would be better: Currently, I am taking the IELTS training and planning to move to Canada by May this year on student visa.

It would sound better if you said "Currently I am taking IELTS training and I am planning on moving to Canada in May on my student visa."

“Currently, I am taking IELTS training and planning to move to Canada on a student visa by May of this year."

“I can currently taking IELTS training, and plan on moving to Canada this year by May—via student visa.”