I used to have dinner at restaurant, but I have started cooking recently. If Simon asked me "how are you lately?", can I say "I am fine. I have started cooking recently."?
Note that we don't say "How are you recently?" The present tense does not go with "recently". The question would either be "How are you?" or "How have you been recently?"
"I have started cooking recently" also sounds strange. If someone said "I have started cooking ..." I would assume they were going to talk about one instance of making a meal, so perhaps they had started cooking tonight's dinner a few minutes ago. A better exchange would be, for example:
Simon: What have you been up to recently?
Me: I've taken up a new hobby - cookery! I've been cooking everything from scratch at home instead of eating at restaurants.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.