IMHO your best bet is to do these as part of the same excursion from your hotel, and it’s probably best to leave it until Saturday (unless you have tons of energy on Friday when you arrive).
If you don’t have too much luggage, the best way of getting to SF from the airport (I’m assuming you’re flying into SFO) is via BART. Get off at the Powell St Station and walk 3 blocks up Stockton (gentle uphill slope), and you’re at the Grand Hyatt.
There’s plenty of stuff to see and do within a few minutes’ walk of your hotel, especially if you don’t mind hills. Union Square, the Tenderloin, Chinatown, Nob Hill, Russian Hill, Telegraph Hill, and Fisherman’s Wharf are all within the “pizza slice” defined by Market St and Van Ness Ave (the edges of the slice) and the bayfront between the Wharf and the Ferry Building (the “crust”). You can also cross Market St for the bars and clubs of SOMA (South of Market). All of the above are within walking distance for a fit person.
My recommendation for the Golden Gate Bridge walk is as follows: take the Golden Gate Transit bus #10 (click on “To Marin City - Northbound (WEEKENDS AND HOLIDAYS)” for the schedule if you’re doing it on the Saturday). This starts out at 7th & Market on the weekends, which is about 1.2km from the Grand Hyatt. Pay $3.25 into the farebox (use exact change - dollar bills are accepted). This bus will take you out to, and across, the Golden Gate Bridge. Get off at the first stop once you’ve crossed the bridge (ring the bell and walk up front just as you’re leaving the bridge: you may be the only person getting off at that stop, since most tourists don’t know about this option). Walk back a couple of hundred meters, and you’re at Vista Point (good for photos of SF). Then walk back across the GG Bridge (you’ll be on the east side, with views of Alcatraz, SF, Oakland, and Berkeley). Once across the bridge, keep to the left and walk downhill at the earliest possible point, and you’ll end up on SF’s northern waterfront at Crissy Field, with continuous spectacular views of the Bay, Alcatraz, and Angel Island. Looking east, you’ll see the Beaux Arts dome of the Palace of Fine Arts, which is right next to the Exploratorium, about a 2km walk away.
[By taking the bus outbound and walking back, you save time by not having to retrace your steps, and you don’t have to wait in a remote location for a bus back towards your hotel. A fit person ought to be able to walk back from the northern end of the GG Bridge to the Grand Hyatt, but if you get tired there are plenty of transportation options once you’re back closer to downtown. If you start out walking to the GG Bridge, you’re more likely to run out of steam in an area where you have fewer public transit options.]
Once you’ve done the Exploratorium, you can walk all the way round the waterfront (Fort Mason, Fisherman’s Wharf for the boat to Alcatraz that you’ll have pre-booked, then Pier 39, then around Embarcadero to the Ferry Building) at water level, or head up from the Wharf via Columbus Avenue (North Beach), to Grant Avenue and Chinatown, on your way back to your hotel.
You’re probably already aware that for the things that you’ve expressed an interest in, renting a car is counterproductive. SF has one of the best transit systems in North America (although there’s a Balkanaization between different transit agencies, which is why you’ll need a different ticket to cross the Golden Gate Bridge from the ones that get you around the City).
If you think that you’ll want to be doing a lot of transit-riding (especially cable cars, which can get expensive at $5 per trip), you might want to look into a MUNI 1-day Passport for $11 for the Saturday. Note that this won’t work on the GGT bus #10 I linked to above, nor get you between SF and the airport. The MUNI bus map is here (warning: PDF).
Keep posting with more details as to your likes and dislikes; depending on your interests and hill-walking abilities, I’m sure that betwen now and March 31 we can come up with 34 hours straight of non-stop San Francisco-y goodness for you!