“Come on! You've got to at least give it a shot.”
There's an art competition that you were thinking about entering, but now you're thinking you won't enter it because you don't have a chance of winning. You tell your friend that you're not going to enter, but she disagrees. She says this to you.
Come on! You've got to at least give it a shot.
Come on! Tell me what she said!
Come on, stay for one more round.
You can also say "Come on!" when a person does something that you don't like, and you want them to stop. For example, if you and your partner were teasing each other, and he gets angry at you, you can say:
Come on! I was only joking.
When you want someone to do a something, but they don't want to do it, you state a lower minimum amount that you think they should do using "at least ___":
You can at least take out the garbage, since I cooked and washed all the dishes for you.
"Give it a shot" means "try it". It's a more casual phrase than "try it", so it's good for conversations with friends and family.