I would highly recommend Sesame Player (Infinite was in season 2) and it's definitely worth a watch. The boys are funny there, you get to see who they are as people, their hardships and how close they are. B1A4 is on season 3 of the show. They're good too but I found Infinite's funnier. MBLAQ was on season 1.
There's Explorers of the Human Body which has EVERY (yes, all 13) members of Super Junior present. DBSK came on two of the episodes.
There's also Super Adonis Camp which was a nice and funny watch too. During that time, they were Super Junior05.
I like Super Junior Mini Drama too. They played some games on the show and it's really nice to see their mini dramas. A lot of them made me laugh but a few made me want to cry. There's also DBSK's Banjun Drama.
I think Strong Heart is a good show too. There are different guests each time and there are different stories being told. You'll see Boom Acedemy (Boom, Leeteuk, Eunhyuk, Shindong) often. It's a heartwarming and funny show.
I think Birth of a Family is a cute one. It has Infinite and A Pink on season 1. In that show, both of the groups took care of abandoned puppies and raised them (Infinite and A Pink were not together). By broadcasting the show, they're hoping that the puppies will get new owners who'd like to adopt and love them. 4minute's Hyuna, G.NA and Z:EA's Siwan are on season 2.
I recommend Wild Bunny too. Those were during the days when Jay was in 2PM and I gotta say, it's hilarious.
I recommend Intimate Note too. Super Junior was on as a guest once and they were hilarious + it really did get them to become closer. The Wonder Girls and Big Bang were on it before. In the season after that, SNSD came.
You should definitely give Family Outing a try. Though, I recommend the first season since that one was a success and they were all close + funny. DBSK's Yunho and Junsu was on an episode.
Come to Play and Happy Together are great shows as well! Jaesuk's a good MC and there are funny stories + touching ones they share. There are different guests too.
Midnight Idols a good show as well. Infinite was on it before and it was another funny one.
Oh! We Got Married is recommended as well. It's when celebrities go on the show, get paired up and they act like a married couple. It shows the viewers how it'd be like if they really were married and in fact, fans actually end up encouraging their star to date their on-screen partner or just date in general. The couples that have featured before were 2AM's Jokwon and Brown Eyed Girls' Gain, SNSD's Seohyun and CN Blue's Yonghwa, f(x)'s Victoria and 2PM's Nichkun. Now, Super Junior's Leeteuk is on it with actress Kang Sora (she's in Dream High 2).
I don't think this show goes on anymore but it's call My Teacher is Coming. Different celebrities have been on the show and it's really touching to see them reunite with their teachers.
For the music programs, Inkigayo and Music on Top are there too.
Hope I helped. :)