I hardly ever feel the need to use a semicolon. I think this particular punctuation mark muddies the page (visually) and obscures the relation between ideas. You can get by pretty well with periods and commas, with colons used to introduce quotes. The semicolon is used only to connect two short sentences that really can't stand on their own, or to separate items on a list when these items are phrases with interior commas rather than single words. In the first case, I just write two separate sentences. If they are choppy, then I recast one of them, or combine them into a single sentence.
I reviewed an article the other day for a nameless journal. The article was written in Spanish and the author had the most annoying habit of separating the subject and the verb of a sentence with a comma. Writers of Spanish do this sometimes when the subject is a longer one, but it is not correct usage.
You usually don't need a comma after thus or and at the beginning of a sentence.