When is the best time of the day to learn? 

That depends on whether you are an old adult or a young person.

Older adults have morning brains.[1] The best time for them to learn is from8.30 – 10.30 am.

Researchers have identified more than 100 circadian rhythms that recur daily (Mayo Clinic 1995).[2]  One example is a study on a group of people aged between 60 – 82 years at the Rotman Research Institute at Baycrest Health Sciences in Toronto (Canada).[3]  It noted participants in their experiment were better able to focus and ignore distractions, and did better at memory tests, between 8.30 - 10.30 am than between 1 pm and 5 pm.[4]  fMRI scans revealed the brains of these people were “idling” in the afternoon. They had switched tothe so- called default mode, associated with daydreaming. [5]

This means older adults should try to schedule their most mentally-challenging tasks in the morning. Those include doing taxes, taking a test (such as a driver's license renewal), seeing a doctor about a new condition, or cooking an unfamiliar recipe.[6]

Younger adults, by contrast, are still very active right into the afternoon in brain areas related to the control of attention.[7] Studies show their memories function best at about 3 pm.[8]

A more recent study found that 16 to 17-year-old girls performed better on tests of factual memory if they studied the material at 3 pm rather than at 9 pm.[9]They can better learn movement skills in the evening.[10] 

“The results suggest it might be better to use the afternoon for studying languages, and the late evening for playing piano or another musical instrument,” says Christoph Nissen at the University of Freiburg in Germany.[11]

As for kidsmany believe that morning is a better time for learning for young children. However, a study done by the Center for Evaluation and Monitoring in England found that children learned more in the afternoon than in the morning.[12]
Overview For old people, it is arguable that this trivial knowledge is more for academic interest. However, for young people living in a highly competitive world, it may be useful to know when is the best time to study/learn something. This little bit of knowledge is like being the first to make a move in a chess game. It can give you an edge over the competitor.

NOTE: I have not defined what's 'old'. It is said old age is a 'social construct' rather than a 'biological stage.'  And Mark Twain says, "Age is an issue of mind over matter; if you don't mind, it doesn't matter." However, for those wanting to know how old is 'old', it may be the best  indicator is when you can laugh, cough and pee at the same time.

Gim Teh/05 June 2015

[1] http://research.clps.brown.edu/songlab/Documents/NewScientist.pdf"Older adults have 'morning brains': Noticeable differences in brain function across the day." ScienceDaily. ScienceDaily, 6 August 2014. www.sciencedaily.com/releases/2014/08/140806125100.htm .
[2] Perhaps the most familiar is the sleeping-waking pattern. Less well known is the variation in blood pressure, respiration rate, and temperature over a 24-hour cycle. On average, the human body temperature varies from just above 97 degrees Fahrenheit to nearly 99 degrees (Colquhoun 1971). The lowest temperature typically occurs at about 5 a.m.; the highest, at about 6 p.m.:http://www.ascd.org/publications/educational-leadership/dec97/vol55/num04/Giving-Students-the-(Right)-Time-of-Day.aspx
[4] “Older adults during the afternoon sessions were 10 percent more likely to pay attention to the distracting information than younger adults, who were able to successfully focus and block out the superfluous information”:http://www.latinpost.com/articles/18815/20140807/study-shows-older-adults-are-morning-people.htm
[5] The fMRI data confirmed that older adults showed substantially less engagement of the attentional control areas of the brain compared to younger adults. Indeed, older adults tested in the afternoon were “idling” – showing activations in the default mode (a set of regions that come online primarily when a person is resting or thinking about nothing in particular) indicating that perhaps they were having great difficulty focusing. When a person is fully engaged with focusing, resting state activations are suppressed”: http://www.baycrest.org/research-news/older-adults-have-morning-brains-study-shows-noticeable-differences-in-brain-function-across-the-day/ 
[7] “Older adults have 'morning brains': Noticeable differences in brain function across the day”,
[8] “In 1977, a study found that children's short- and long-term memory varied significantly with the time of day. Schoolchildren taught at about 3 p.m. retained material longer than those taught in the morning. Moreover, seven days later, those who read a story in the afternoon could recall more details than could students who had read it at 9 in the morning: Folkard, S., T. Monk, R. Bradbury, and J. Rosenthal. (1977). "Time-of-Day Effects in Schoolchildren's Immediate and Delayed Recall of Meaningful Material." British Journal of Psychology 68, 1: 45-50. In 1985, a research team working for NASA's Biological Research Division observed that athletic teams instructed in the morning were able to remember the moves about as well as a person who had slept only three hours the night before. As a result, the biologists recommended that instruction in intricate maneuvers be scheduled for the afternoon”: Winget, C., C. DeRoshia, and D. Holley. (1985). "Circadian Rhythms and Athletic Performance." Medicine and Science in Sports and Exercise 17, 5: 498-516; see summary in http://www.ascd.org/publications/educational-leadership/dec97/vol55/num04/Giving-Students-the-(Right)-Time-of-Day.aspx
[9] Timing is everything: Age differences in the cognitive control network are modulated by time of day. By Anderson, John A. E.; Campbell, Karen L.; Amer, Tarek; Grady, Cheryl L.; Hasher, Lynn. Psychology and Aging, Vol 29(3), Sep 2014, 648-657.
[10] John A. E. Anderson,  above, in Psychology and Aging, 2014; DOI:10.1037/a0037243
[11] . “Why should timing matter? We know that sleeping after learning a new factor skill helps consolidate memories” says Christoph Nissen, a ‘sleep specialist’. Nissen suspects that the “critical window” between learning and sleep is shorter for movement-related learning than for other types of memory. Whether adults can benefit as much as teenagers from these windows isn’t clear. “There is evidence that adolescents have a higher capacity to learn – and they sleep better,” he says.Who is Christoph Nissen?  He was Executive medical supervisor medical director of the sleep laboratory, research group leader ‘Sleep and plasticity’, representative for teaching. From 2001–2004, he was resident and research associate at the Department of Psychiatry and Psychotherapy, University Medical Center Freiburg. From 2005–2006 he was research associate in the Sleep and Affective Neuroscience Laboratory at the Western Psychiatric Institute and Clinic, Pittsburgh, PA.  In 2000 his dissertation was awarded summa cum laude by the Department of Psychiatry in Freiburg.

[12]  Eleven out of twelve tests slightly favored afternoon learning. The differences were not very large but do point to the fact of only small differences between learning in the afternoon as compared to the morning”:  http://everydaylife.globalpost.com/morning-afternoon-optimal-learning-time-kindergarten-16702.html