Dialogue has always been a strong point for me. That’s why plays are easy for me to write. I have an ear for how people talk and I can transfer that to paper. Now, I struggle with description, which we’ll tackle another day. You may be the opposite of me. So, how can you get better at dialogue? Here are some pointers:
Listen - When talking with people or in a crowded place, listen to people and how they talk. How do they put their words together, phrases? You may even write some of it down. Once story I remember, now whether or not it’s true is beyond me, is about Quentin Terintino. Before he made Pulp Fiction, he sat in a bathroom stall and wrote down a conversation he heard going on. From that conversation came his most famous movie.
Transcribe - I talked about this above. You have to sit down and write the conversations. One thing you can do is record a conversation, then transcribe it in story form. No, it may not make for a good story, but this is just for practice.
Practice - Practice. Practice. Practice. You only get better at something by doing it. If you know this is your weak spot, do more and learn more. You may never be a playwright, but you will be a better write in your own right.
Any good writing resource can give you the rules for using quotation marks. If you are a writer, you should already have at least one of these reference books in your writing space.