I will echo Shannon. I have a 10-yr-old boy in 5th grade and a 6.5-yr-old girl in 1st grade. Both are in new schools this year. He is in a private school and she is in a public school. I never used to initiate playdates for him when he was younger (my mistake) so deep friendships didn't really develop the 4 years in was in a public elementary school. He was sad that friends never called him for playdates and I was surprised he wasn't invited to birthday parties. It turned out, what I needed to really do, was initiate it so friendships can develop.
BTW, he does have friends on the block, but this is hit and miss. Not everyone is so nice to him. So I actually prefer he doesn't play with them anymore (or as much as he used to) and deepen bonds with friends he really connects with at other places.
Anyway, it's been really hard for me personally to pick up the phone and initiate, but that's what I've done and it's made a huge difference. Since December, my son really felt better about his new school and this was simply due to feeling connected to 2 kids out of 24. I asked him who he wanted a playdate w/ and he gave me 3 names. So we take turns, he goes over there for playdates/sleepovers or they come here. It's nice because now the mom feels comfortable calling me to have her son come over (he was begging to play with my son) vs. staying home with a sitter. It's a win-win.
My daughter also has been making friends, but again I call to initiate and then people feel comfortable calling her up for a playdate. I don't call the same child over and over again, I try to mix it up so she can feel comfortable and happy with a wider group (now it's 3 girls).
When she was an infant, I had a very good girlfriend and we hung out a lot and now both our girls are Best Friends. They love each other and I'm so happy for that. She goes to another school, but they still ask for each other. Our families know each other so well, sleepovers are not out of the question. So I don't think this is too early. In fact, I believe if you wait for middle school for them to find friends, it will be a lot harder for you to get to know the families.
My daughter stays in her room and plays with her friend.
My son is more active and prefers me to take him somewhere fun with the friend and I'm happy to do that on occasion. Or they stay home (we have a huge park across the street). At night they watch a movie for a sleepover.
I hear that more from my son, who is very social and friendly and outgoing. He is extroverted so he definitely needs "friend" time. Introverts crave alone time. I haven't done the "here is the list of chores if you are bored" but I have begun doing it recently.
I used to do the same thing. Now I try to focus on a few close friends they want to hang with. You really can't depend on one day for friendship to be formed. It has to be one-on-one.
ACTIVITIES & HOMEWORK
If they are exhausted when they come home, you can't really load activities on top of them. Do they have outside interests? If they do, then they will ask for the activities. One way to get them interested in activities is to have them do it. You need to take them out and introduce them to it.
Try to be in a good mood. Imagine someone videotaping you after school and see if you like what you see. If you don't, change your attitude and lighten it up. I have recently turned into a harpie at bedtime because I was exhausted at the end of the day and they were dawdling. Now I work hard on my patience and keeping it loving and kind.