Hi! Welcome to the original Sid Meier's Civilization!
When engaged in a war against a civilization, it is sometimes difficult to obtain peace with that same civilization. For obtaining peace, you need to talk with that civilization leader. You can wait for them to contact you (contact is always made via units; try to move one of your units to a square adjacent to an enemy unit) or you can take the initiative by moving a diplomat unit to any of their cities (it needs to reach a city, reaching a unit won't do) and selecting the option "meet with king".
However, bear in mind that it may be hard to move your diplomat through the enemy's country. The diplomat is a powerful unit (maybe the most powerful of all) but it is very vulnerable to attacks from enemy military units.
Besides that, the enemy civilization may not be interested in making peace with your civilization unless you give them technology and/or money. In some cases (for example when you don't possess any technology that they don't have and your treasury is almost empty or when they feel that you are a lot weaker than them), there is no way for obtaining peace. You must continue the war.
You should also know that there are two wonders of the world which force the other civilizations to always accept peace with the civilization that holds any of them: Great Wall and United Nations. So, you can try to build one of these for being able to obtain peace easily. However, even by possessing one of these wonders, you always need to talk with the other leader, i.e. completing the production of Great Wall/United Nations doesn't automatically stop any war.
Finally, let me congratulate you for trying to play peaceful games. In my opinion (and I'm not alone) it is much funnier (and harder) to play for peace than "total war" style.
