Renting a car, so travel is not a problem, as long as within a 2 hour drive. Group of 5, and working on a moderate budget for places to stay.
I recommend Muir Woods for a natural park to visit. It's located just north of San Francisco, across the Golden Gate Bridge and you get some beautiful views of the bay and the city. If you want to go a little further north, Point Reyes National Seashore is another great park to visit.
In terms of places to stay, I would suggest taking a look at places on AirBnB, since you might be able to find a nice 2 bedroom apt to rent for your stay. In terms of hotels, you can find some decent deals on motels in the Fisherman's Wharf area and they're located in a great area for tourists, with Aquarium of the Bay, Ghirardelli Square Shopping Center and many other attractions and restaurants.
Have a great time and have fun exploring the SF area.
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Be careful with renting a car. It is hard to find parking and you might get stuck paying more for overnight parking than you did renting the car! Public transportation in SF will get you within 2 blocks of any address in the entire city.
For places to stay in San Francisco for a group of five with a car on a moderate budget, I would stay out of the downtown and other touristy areas because it's a huge hassle to find street parking and parking lots are expensive. I would go to Airbnb.com and look for a house that can accommodate for 5 and these are the neighborhoods I would suggest: Mission District, SOMA, Richmond District, Sunset District, and Haight-Ashbury. You'll have much better chances of finding a place big enough for five that also has parking in those neighborhoods.
For nature-y things to do, this is a little further than 2 hours, maybe 2.5-3 hours drive, but it's completely worth it -- Big Sur. Just type Julia Pfeiffer Burns State Park into your GPS, and you can also pull over on turnouts along the road. So gorgeous!!!!! On the way to and from, you can stop off at Half Moon Bay, Santa Cruz, and Monterey for beautiful beaches and attractions.
Have fun and let me know if you want any other suggestions! :)
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Hi there is a great huge park in the center Golden Gate Park and there are many museums there on the first Wednesday of every month they are free, and on Sundays they close the park to cars so it is great to go rollerblading or biking.
If you've got a car, you'll find it cheaper to stay OUTSIDE of SF -- the parking alone will cost more than a moderate budget is comfortable spending. And street parking is near impossible to find, not safe, and you'll probably end up with a ticket for something you did wrong.
Solution: stay at a hotel just outside of SF. There are a great number of places not far from the airport (which puts you 5-15 minutes away from the city - as long as you don't drive during high traffic time). Look around Oyster Point Cove -- parking is free and it's close to the city.
The others have made great driving/day trip suggestions.