The verb that comes to mind that describes this action is wander, which means to go about from place to place usually without a plan or definite purpose, and can be used to describe a situation where one walks around slowly in a relaxed way or without any clear purpose or direction.
We spent the morning wandering around the old part of the city.
Another possibility is stroll, which, in a similar fashion to wander, means to walk in a slow, relaxed manner, especially for pleasure.
I strolled around the old part of the city.
I went for a stroll in the park.
If you wish to expand upon stroll, and reenforce the point that your walk in the park is done in a relaxed way, without hurrying, then leisurely can be used as an adverb in this case, and indeed, is often used in conjunction with stroll to describe a relaxed and unhurried walk somewhere.
I took a leisurely stroll in the park this morning.