Do I have to pay anything if I trade in my iPhone 11 for a newer iPhone?

I see no one has provided a link to show you how to trade your iPhone and what value it may hold. To simplify, you won't be able to hand Apple your iPhone 11 and walk out of the store with a new iPhone without paying the difference between whatever Apple gives you in Trade value.

As others have pointed out, your cellular provider may have trade in options which will give you greater credit for your phone. But this link will tell you how much Apple will give (assuming there is no significant damage or issues with your iPhone 11) --> Apple Trade In - Apple

You would need to look at the offers for whatever carrier you use to see what your possible savings could be. Or if there is no carrier offer then you will get whatever amount quoted based on model and condition. You will likely pay something. Just can’t tell you how much

If the OP is in the US and uses one of the carriers (Verizon, T-Mobile/Sprint, AT&T) they could qualify for $800 if they choose installments. They haven’t really provided any details to help give a real answer or even guesses.

You will have to pay

An iPhone 11 has very little resale value at this point

About $220 if you trade it in, $350 if sold privately.

If you are going to a physical Apple Store to buy your new iPhone you just take the old one with your and trade it in there and leave with your new one. This assumes you are not making payments on your iPhone 11.