My favorite subject is definitely science. My favorite subject is science.

I especially liked physics when I was student, but I likelately I've been enjoying biology as well lately. I especially liked physics when I was student, but lately I've been enjoying biology as well.

I rephrased the second half of the sentence so that it sounds more natural. I feel "lately" is most often used with actions, so the original sentence sounded a little odd to me.

I especially liked physics when I was a student, but lately, I like biology as well lately. I especially liked physics when I was a student, but lately, I like biology as well.

I especially liked physics when I was student, but I like biology as well latelymore recently I am liking biology too. I especially liked physics when I was student, but more recently I am liking biology too.

I've started to find outdiscovering the beauty of the nature since I paintedbegun making oil paintings. I've started discovering the beauty of nature since I begun making oil paintings.

(1) I think the use of "discovering" here would sound more fluent and perhaps a little more appropriate. (2) Unless you're referring to the nature in a particular place, we will not use "the" before "nature". (3) I also rephrased the second half of this sentence to sound more natural. (4) I replaced "paint" with "make" because "painting oil paintings" might sound a little silly.

I've started to find out the beauty of the nature since I started painteding oil paintings. I've started to find out the beauty of nature since I started painting oil paintings.

Since I started painting oil paintings, I've started to find out the beauty of the nature since I painted oil paintings. Since I started painting oil paintings, I've started to find the beauty of nature.

I would like to see the various types of animals all around the world. I would like to see the various types of animals all around the world.