It's sometimes called "telegraphic style" - as if you were sending a telegram and paying for each word, you would leave out some words to make it shorter. You might write like this in a diary, or in a very informal letter, especially if almost every sentence would otherwidse begin with "I."
In the other example you give, "I like to write, surf the net, and am also a member of the International Club," you need the "am" because what follows it is not a construction parallel to "I like to write" and "(I like to) surf the net." It's not a great sentence either way - I wouldn't say it's incorrrect (with "am") but it seems to me that in a list of three things they should be parallel in construction. I would prefer either a more parallel construction : "I like to write, surf the net and attend the International Club" or else two more distinct clauses: "I like to write and surf the net, and I am also a member of the International Club." (without a comma after "write" it looks less like a list.) I'm not sure what the appropriate grammar rules here are, I'm just trying to explain why the original sentence sounds rather awkward to me.