Hoo-boy have you hit on a hot grammar topic.
The word however is a conjunctive adverb. This fancy title simply means you can use it as a transitional word between clauses or as an adverb modifying a clause.
Either way, its basic function is to provide meaning to the relationship between sentences or clauses. Other examples include therefore and nevertheless, or henceforth and heretofore (if you like sounding like Shakespeare).
These words are flexible, so… yes… it's fine to use them at the beginning of sentences. (Don't hate us, Internet users who disagree.) You just need to recognize when you need a comma. If you don't use one, however means in whatever manner or to whatever extent.
"However I liked the lasagna Aunt Miriam made last night; normally I cannot even stomach the thought of cheese."
"However, I will buy you another set of Pokémon cards if you promise to eat all of your eggplant tonight."
Which sentence is correct?
It's definitely not the first one.
That speaker needs to spend less time chowing down on Italian food and more time digesting grammar. In fact, their sentence actually has two technical errors. First, without a comma, this sentence basically means, "In whatever manner I liked the lasagna…normally I cannot even stomach the thought of cheese." That doesn't sound right.
Second, using a semicolon indicates that the first part of the sentence is an independent clause. Again, the lack of a comma negates that possibility. What's the solution? Put a comma after however. Big problem; easy fix. We love it when that happens.
The second sentence uses however correctly because it combines the conjunctive adverb with a comma and means "In spite of whatever heinous thing you did earlier, kiddo, I will buy you some Pokémon cards if you start wolfing down that eggplant."