"Chance" is standard English and the safe choice for a learner. "Shot" only means "chance" in certain contexts, and it is informal English. Suppose someone doesn't think they can do something. You want them to try, anyway. "Go ahead, take a chance" is standard English. "Go ahead, take a shot at it" is informal, conversational, spoken English. "Take a shot at it" suggests that success is unlikely, but possible. It also suggests that you think the experience itself is worthwhile. "I can't possibly win the race, so I'm not going to enter." "Go ahead, take a shot at it. You won't know until you try. And it is good practice." Also, "take a shot at it" means you are going to do something, try something. "Take a chance" can simply refer to making a choice involving luck. "During this intermission, we're having a fifty-fifty raffle to support our theatre club. People are in the aisles now, selling tickets. Come on! Buy a ticket! Take a chance! You can't win if you don't play!" "Even though it's longer, I'm going to take Route 24 instead of I-95. I can't remember if there's a football game on today, and I don't want to take a chance of getting caught in football traffic at Foxboro."

They can be used interchangeably in this sense. I think "shot" may be more common in American English and "chance" is probably more common in British English.