Firstly, if you are going there with zero dollars you can enjoy...people watching? There are trails, grass fields and picnic areas covering the whole place so you can enjoy just being outside and whoever you are with. If that certain someone is kind of boring and you still don't have any coins there are free guided tours at the San Francisco Botanical Garden . But if you are not fifty yet and would like a little more action in your day, the disc golf course would do you right. Come equipped with the discs and the course is free.
If you want to leave the park a little more cultured, you have some things to choose from on little cost. Firstly, the Japanses Tea Garden is pure serenity in the crowded city. If I remember correctly it is free if you go in for the last hour or so. I say just pay the $5 so you have the time to sink into the pace of the gardens. It is well worth it. Right next door is the De Young Museum It is free if you go on the first Tuesday of the month. If not its about 10 bucks, and you save $2 if you show a Muni pass or transfer. I got there within the last half hour of closing so I got in free and ran through only one of the many halls they have. I don't recommend seeing it in that fashion, but walked passed it and walked in.
So thats that. There is so much more in this park than you could tell from the outside. Just go in a explore the place. Finding new things don't cost a damn thing. However actually doing them might, so tough rocks.