“Tone” is something that worries many students. And here there are no hard and fast rules. Tone and style are subjective, and what is appropriate or inappropriate has varied from period to period. It would once, for example, have been frowned upon to start a sentence with ‘And’, as in the previous sentence, or with ‘But’ for that matter. But it has become acceptable enough.
The reason for this is that writing has become simpler, more direct and more conversational. Victorian writing was very different, for instance. The Victorians were fond of punctuation, but we tend nowadays to remove unnecessary punctuation to aid the readability of our writing: as a small example, to write ‘the 1600’s’ is perfectly acceptable, but so too is ‘the 1600s’, and for simplicity’s sake the latter is preferable. A good rule of thumb is that the writing should flow naturally, and not appear forced. We can use ‘And’ or ‘But’ to begin a sentence because the meaning is readily understood and it does not ‘feel’ wrong. Read a sentence over, and if in doubt, simplify rather than over-elaborate.
A similar argument can be made for the use of ‘I’, the personal pronoun. You should not overuse this, as it can make essays sound too personal and not professional enough, but there is no absolute ban on using the first person in your essays. It can be very effective, again for its direct address to the reader. There is no need to say things like ‘In my opinion’ (for whose else would it be?), but saying things like ‘From my background as a white, middle-class European man, I would certainly know more about the history of Europe than of Asia, and am thus more susceptible to Eurocentric ideology. I grew up with little enough knowledge of Europe’s history, but with virtually no knowledge of world history’ is perfectly all right. It makes a fair point.
If you take this too far, however, you do risk sounding rather too casual. For this reason, you should not in an essay use idiom and colloquialisms – slang or everyday words. You should also use full forms, not contractions: ‘do not’, not ‘don’t’, and so on.