S.C.
The 1st Birthday is really more for the adults than it is for the birthday boy or girl.
We have all of our kids birthday parties at our house. I never really plan any entertainment. The kids all just play and the adults sit around and chat. I always serve lunch, then we do presents, then the cake.
I think for my daughter's 1st we might have done the cake then the presents, but I'm not for sure on that one.
For the 1st Birthday you'll want a smash caked for the birthday kid. They get to just tear into it with their hands. You can get the really small cakes from the bakery, just tell them you want a small smash cake and they'll know what you want. Make sure to strip your little one down to their diaper for that part! And get the camera ready! :)
I've never done party favors for anyone. Most of the time party favors are just junk anyway. We've received a number of party favor bags and the toys go straight in the trash when we get home. If you do decide that you want party favors, do things that kids will use. Stickers, bubbles, pencils, small activity or coloring books are all good ideas.
As far as the food, it depends on what time you have the party. Like I said, I usually do a lunch. I've had a few different things.
I've done sandwiches. I buy lots of lunch meat and put it on trays. I usually make up a few cheese and veggie trays as well. If I do sandwiches, I almost always to do chips with different warm dips (buffalo chicken, spinach artichoke) and some salsa.
Last time I had a party, I ordered a bunch of pizzas. It was about the same price as the other stuff and a LOT less work for me! :)
My youngest daughter has a July birthday so we will have a cook out. We'll have hot dogs, burgers, brats, potato salad, pasta salad and chips.
If you want to just have cake and ice cream, plan the party for around 2:00 and make sure you put on the invite "Join us for cake and ice cream!" I would still offer a few munchies though.
Have fun! :)