When using but and yet as coordinating conjunctions, you can only use one at a time (otherwise, you’re creating a redundancy for the category of the redundancy category :)).
Use either but or yet when conveying two ideas that are in contrast to each other in order to separate them. His family lives in Tampa, but he lives in Iowa or His family lives in Tampa, yet he lives in Iowa. Do NOT say His family lives in Tampa, but yet he lives in Iowa. That is a redundancy.
Use and when conveying two ideas that are connected to each other. The example above is not an appropriate place to use and because the clause before the comma (His family lives in Tampa) and the clause after the comma (he lives in Iowa) are a contrast, not a similarity. A better example would be His family lives in Tampa, and he lives in Tampa.
Don’t use and and yet together when both are being used as coordinating conjunctions because they convey opposite ideas. As a coordinating conjunction, yet means nevertheless or however, and and conveys a meaning of in addition. Therefore, His family lives in Tampa, and yet he lives in Iowa makes no sense. It says His family lives in Tampa, nevertheless, in addition, he lives in Iowa.
I know, I know, someone out there is saying, But wait (or But yet wait); what about Shakespeare? He used but yet. Well, no offense to the bard, but Shakespeare was a famed poet and playwright—not a famed grammarian.
Paul’s book–Literally, the Best Language Book Ever and Sherry’s Grammar List