To enter a challenge, go to the Explore tab. Here, you can find any and all new, live and soon to be released challenges. Depending on your trainer, there may be a number of challenge variants that you can pick from - have a look through the descriptions for all of them, and then select the challenge that you feels most suits you.
Once you’re in a challenge, you can technically switch to another one - but you cannot join it halfway through. That’s the point, it’s a challenge! So remember, it’s important to pick well at the start.