If you don't want to start sentences with conjunctions, you are free to do so.
A simple way to do it would be simply to stop using and, but, and or to connect clauses or phrases. Connecting words, between two words, or before the last of a list of words, OK:
- I saw Bill and Bob at the picnic.
- I saw Bill, Bob, Ben, Mary, and Sarah at the picnic.
- I saw Bill and Jen saw Bob but they were both with somebody else, or maybe they were alone, and ....
- He arrived on time, but then he went back to the car and got a gun, or talked to somebody, and ...
There's nothing wrong with these sentences, mind you. They're perfectly normal, the way people talk all the time. But if that's the way you talk, you probably don't want to write that way; as you say, it goes on and on. And it's good practice to try something else.
So take your time and try to become aware of conjoined clauses. A good exercise is to write each clause as its own sentence. Then figure out how they should be arranged to get your ideas across. Without conjunctions.