"What are the best times of day to do archery?"
Obviously shooting during the day is good so you can see the targets better. Speaking for myself, I find the early mornings are tricky because the sun is sometimes in your eyes. As such I prefer to do archery between 10 AM and 6 PM.
I also have a fondness for doing some night shooting in the hours after sunset - in which case it is useful to have lighted LED nocks, like the ones shown below. I also recommend bug spray, as depending on the time of year there will be lots of mosquitoes after dark.
|Lighted LED Nocks for Night Shooting|
Some people also prefer to shoot on weekdays, when there are less people at the local archery range, while others might prefer weekends or the "after 5 PM crowd" because they like to socialize with their fellow archers. Archery can be a very social activity. Not everyone is into socializing, so it really depends on your personal preferences.
Some people really like shooting in the early wee hours (5 AM to 7 AM) so they can get some shooting in before going to work. Not my thing personally. I like sleeping.
Regarding time of year...
My favourite time of year for archery is September and October, because the weather is so nice. Winter is cold and if you are a bad shot you could lose arrows in the snow. Spring is very muddy and rains often. Summer is too hot. Thus late summer / early autumn is a great time of year, weather wise.
Depending where you live in the world the seasons may vary dramatically, such as rainy seasons / dry seasons, and so forth. Whether you like shooting in the rain is a whole different topic.
How to do Archery in the Wind and Rain, Part 1
How to do Archery in the Wind and Rain, Part 2
Sunrise, Sunset and Daily Weather Forecasts