It's easy enough for me to go on the UESP and find the answers, but I'm into roleplaying and was wondering if and how my character can find out what days to summon certain Daedra Lords in-game. Also I was wondering if there's any in-game evidence of where the witch covens are, or if the character has to just stumble upon them?

I think, but am not sure, you stumble on a few covens en route to other places whilst doing quests. I could be mistaken though, I'm not that far into the game.

As for the summoning days....as a person from that world, you should know it's own holidays. Many Daedra have summoning days on holidays I think. Again, little experience, so I could be wrong.

There is an in-game book in which all the summoning dates are listed. I won't say which unless you ask, because being completely spoiled on where to find the information in-game is really the same thing as just looking up the summoning dates online

If you cave and don't want to figure them out for yourself, you can find a chart http://www.uesp.net/wiki/Daggerfall%3a%44aedra_Summoning#Summoning_Days on the UESP Wiki.

I don't know if the dates are ever actually stated in-game. However, when arriving in towns on holidays, you get a status message about the date and what special event (free temple healing/reduced spell prices/free ale in taverns) is going on. Something similar may occur on the Princes' summoning days, but I'm not sure.