"I am beginning". I cannot tell you why this is the only correct form, but I know that it is the only correct form here.

1. At first I didn't like my job very much, but I'm beginning to enjoy it now. The adverb 'now' is added for emphasis, e.g. I am now beginning to enjoy it. Rather than saying " I only recently began to enjoy it'. English likes the ending -ing. It might be called progressive, or present continuous, or similar (I don't study English). Number 2 (I begin to enjoy it) is an old-fashioned way to say it. Now we prefer to say many things with the -ing ending: I am eating dinner; I am watching a film; I am enjoying work (only now) because I understand my work better (than a few months ago). I am doing: I am studying this language, etc. These words with "ing" endings show that it is happening now, and it is continuous. You can say "i eat dinner, I study xxx language; I watch a film, etc, but then you need a time word - e.g. I eat dinner at 6 o'clock, I watch this film occasionally/rarely/often, I study each day; I enjoy work (i.e. every day). These words without the "ing" imply regular action or that you also need a word to say about the time, e.g. when, how often, etc.