I'm sure you've received lots of amazing ideas, but I thought I'd add a little note about my experiences with having birthday parties for my son. We live in the greater Sacramento area, I don't have a large house, and his birthday is in December. I've done his birthday parties at home (his last one he turned 6). I time it for approximately 2 hours. We start out with some unorganized play time until all children arrive, then we do a craft, after the craft, we do an organized game, then we eat (I plan them Saturdays around lunchtime), we open presents, have cake then the children do more unorganized play. It's during that time that parents get the hint that the party is officially over, and they leave as they want. We've done themes--thomas the train, pirates, superman, etc. Sometimes the craft is actually a snack (I can remember having the kids make their own english muffin pizzas and fed the adults papa murphy's pizza). For guests, the rule of thumb once he turned 5 was one guest for each year--I don't count parents and sibs as the 5 he invites.
This year, my son being in kindergarten has been invited to lots of birthday parties--parents hold them at Bounce U, Whacky Tacky, Sky Zone, Chuck E. Cheese and Incredible John's, but I still did his 6 yo birthday party at home. I imagine next year, we'll choose one of those spots.
Really at 3 years old, you don't have to do too much. Invite family who have children her age or invite children from playgroups or childcare, but you don't have to do too much yet. If your house is too small, is there a family member who will co-host the party with you and allow you to use their house?
From 5 yo on children really get what is going on, and that's when you'll have to get organized.
Whatever you decide, just remember kids love birthday parties because of the cake, so have a good one and enjoy. As the saying goes, they grow up fast.