One reason for saying "well" at the beginning of a sentence is to show that you're unsure about something.
This can happen when someone asks you a question that you don't immediately know the answer to:
A: What are you doing this weekend?
B: Well, let's see... Oh! I'm having lunch with Gavin on Saturday.
It can also happen when you're trying to say something nice, even though you're thinking things that aren't nice.
A: What do you think of Lana?
B: Well... she's... always very punctual.
You shouldn't use "well" in this way in formal written English.