For both my kids' first birthdays we didn't do a whole lot. I made a cake for them and invited some close family members and had a small party. Most of the time the kids were in the high chair trying to cover themselves completely with frosting. I hadn't started making fancy cakes yet so for my son's first i just made a plain 9X13 cake with his mane on it and gave him a piece to dive into. by the time my daughter was turning 1 i had been making fancy cakes for a while so i made her a small one with pretty pink frosting for her to smear on herself.
The parties were very small and ended with a nice bath to wash off the frosting before the kids got too tired and cranky. something to remember is your one-year-old probably isn't nearly as excited about the party as you are and they will tire and get worn out fast.
We had a few gifts and they got to open them before the cake which allowed the grandparents time to play while i got the cake ready. I didn't make a meal for everyone either, just had cake and ice cream so it would stay a small and relatively fast affair.
Now their second birthdays were a bit bigger and fancier, but they were old enough to handle a bigger party.