For informal written language, it's fine. But (!!) if you are doing a more formal piece of writing, you should use 'However + comma' instead.

You can! It is a grammar MYTH that it's incorrect to start a sentence with "but". It's fine, it's correct, it's acceptable, it's in the Chicago Manual of Style. But do keep in mind that it's informal. I wouldn't use it in a cover letter because it's informal and also because chances are that the person reading it might think that it's incorrect.

I too, believe this a grammar myth. And I do not believe it represents a change in usage. Melville, "Moby-Dick, or, the Whale." Within the space of less than a page, he begins a sentence with "But," and then he begins a PARAGRAPH with "But." "...as if striving to get a still better seaward peep. But these are all landsmen; of week days pent up in lath and plaster—tied to counters, nailed to benches, clinched to desks. How then is this? Are the green fields gone? What do they here? But look! here come more crowds..." Dickens, "David Copperfield:" "On the second branch of the question, I will only remark, that unless I ran through that part of my inheritance while I was still a baby, I have not come into it yet. But I do not at all complain of having been kept out of this property..." Addison and Steele, in dedication of "The Spectator" (and surely they would have been careful about their grammar in something addressed to a member of nobility:) "I know that the Homage I now pay You, is offering a kind of Violence to one who is as solicitous to shun Applause, as he is assiduous to deserve it. But, my Lord, this is perhaps the only Particular in which your Prudence will be always disappointed." Shakespeare, "The Tragedy of Hamlet, Prince of Denmark:" "...I know you are no truant. But what is your affair in Elsinore?" Genesis 6:7, King James version: " And the LORD said, I will destroy man whom I have created from the face of the earth; both man, and beast, and the creeping thing, and the fowls of the air; for it repenteth me that I have made them. But Noah found grace in the eyes of the LORD." I am not even going to consult a usage guide. If it's good enough for Melville, Dickens, Addison and Steele, Shakespeare, and the Bible, it's good enough for me.

In grammar school we are taught to never begin a sentence with a conjunction (and, or , but) because it would be considered poor grammar usage.

I have always consider it incorrect to start sentences with 'but' and 'and', but then I've probably done so myself. I such circumstances, it is usually better to new "oppositional" sentence with 'However', 'Nevertheless', 'Be that as it may', etc. In other words, there are better ways to express the same idea.