Is there any way to prevent my tampon from coming out when I poop?

You can definitely try. A few approaches you can take are: a) eating foods that prevent constipation and help make stools easier to pass, b) avoiding bearing down unnecessarily during bowel movements, and c) trying alternatives to tampons, such as a menstrual cups, which are more likely to stay put.

Do I have to change my tampon every time I poop?

Only if you get poop on the string! Feces can contain harmful bacteria and cause vaginal infections if it gets too close. You can hold the string to the front or side to avoid getting feces on it, or tuck it into your lovely labia :) 

Any tips for wiping when dealing with poop and your period?

If you don’t use a tampon or cup, the P+P combo can get especially messy. Flushable wipes are a great thing to keep on-hand for these special moments. Try to find wipes that are free of perfumes and chemicals to avoid drying out or irritating your skin. And if you don’t have wipes, wet toilet paper is a great back up.