One of the hardest places to locate — unless you have directions, or you've been there before — is the Rose Garden. It's toward the west end of the park, a ways from Country Club Road. (More directions below.) Inside a circular fence, you'll find paths between the rose bushes and benches where you can sit — but not much shade other than the central ramada.
It's not open all of the time. For instance, when I went by this evening, the gates were locked: The garden is closed from (this year, at least) January 1 to March 18, waiting for the spring bloom.
Location: Turn east from Country Club at Eastland Street and drive to the farthest parking lot. (Here's a Google Map.)
Hours: The gates are generally open from morning to evening. It's closed from 9:30 to noon Thursdays, and during much of the winter. I like morning hours best... especially during our hot summer months. (To check current hours, try calling the Parks and Recreation office or Reid Park administration. I'll try to get a good number and update this post... or, if you find a good number, please leave a comment below.)