Information is quite scattered, so I thought that my experience could help someone else as well. So, here is an info package on how to travel from San Francisco to Yosemite without a car:
1) Take the BART metro from San Francisco airport to Richmond (9,85 $, about 1h 40 min).
In the metro, you have to change from the yellow line (Pittsburg/Bay Point-SFO) to a red line (Richmond-Millbrae Line) or an orange line (Richmond-Fremont Line). You can do this in many stations, check out the metro map. (On Sundays the red line is not operating, so then you need to transfer at 19th St/Oakland or at MacArthur to the Orange line)
Also, remember that it’s very difficult to get to Yosemite in the same day you arrive to San Francisco (unless you arrive very early), so I’d recommend staying one night in a hostel in San Francisco. There are many hostels very close to BART stops. For example “San Francisco International Hostel" is only 200 meters from Powell station and a four person dorm is 38$/night/person (+$10 in cash for room key deposit). The hostel is pretty shitty and the elevator didn’t work, but it’s probably ok for one night. Then continue early so that you can take the first train/bus combo to Yosemite.
BART website
2) Take the AMTRAK-train from Richmond to Merced (24 $ / 2 h 40 min).
You can buy the ticket from the station, or an e-ticket at AMTRAK website. It’s a good idea to buy a train/bus-combo e-ticket. From Richmond, CA (RIC) to Yosemite, lodge bus stop, CA (YOS) (35 $).
The first train from Richmond to Merced goes at 7:49 am and arrives at 10:31 am. Lines 713, 714 and 716 head to Yosemite.
AMTRAK website
3) Take the YARTS-bus from Merced to Yosemite (13 $ / 2 h 30 min)
The first YARTS-bus after arriving to Merced goes at 11:00 am - 1:25 am. A later connection from Richmond goes at 1:34 pm - 4:13 pm and the bus from Merced goes at 5:30 pm - 8:22 pm.
YARTS website. You can buy the ticket from the bus, or at the AMTRAK website.
In all transfers, you need to carry your bags less than 50 meters.
When coming back, you just do the same other way round. It’s a good idea to book a late flight (departs late in the evening) back home. Then you don’t need to stay in a hostel again. The first YARTS-bus from Yosemite to Merced goes at 10 am, and you will be at the airport around 3:30 pm.
Remember, if using public transport, you will arrive to Yosemite earliest in the mid day, and then it’s very difficult to get a campsite from Camp 4. It’s possible to book campsites at Upper/Lower/North Pines (a campsite for six is 26€/night), and this might be a good backup for the first night. On the other hand, then you need to wake up at 4 am or so and hike to Camp 4 to wait in a line. You should be in the line latest at 6 AM. The first Yosemite shuttle bus starts at 7 AM so it’s not helping.
Until you get a campsite, moving around in Yosemite with all your stuff might be tricky. Everything is much easier if you already have some friends at camp 4. If not, you can ask if you could borrow someone's bear box (You can't leave any food anywhere) or maybe leave some stuff at camp 4. A six-pack of beer might help in negotiations.
Also, at least in 2015 it was ok to sleep in the Camp 4 waiting line, so this is also one solution for the first night. Sleeping in the forest is not allowed, but people do it.
And this is important: normally you can stay up to 30 days in the valley, but in the summer season (May 1 - Sep 15) only seven days are allowed. However, time spent on a bigwall is not included, though you can't leave your tent at Camp 4. To avoid extra hassle, most climbers arrive on Sep 15 or later.
Yosemite shuttle bus operates year round, but El Capitan shuttle operates from May 22 through October 12. So getting to El Cap might be harder after Oct 12 if you don't have a car.
You can find more info at Yosemitepark.com and nps.gov
Getting all the stuff checked in as one piece of luggage might be impossible.
I tried to pimp my belayseat by drilling some holes... Maybe a bit too light & fast.
Finally... British Airlines takes 72€ for the first extra bag (and 150€ for every bag after that...) The 35 liters haulbag was my hand luggage and the laptop bag didn't cause any problems, even though Virgin Atlantic allows only one piece of hand luggage.
The elevator didn't work
Not so sober american tries to lift the haulbag
So, before you decide to travel without a car, make sure you can carry your stuff.
Sleeping in the Camp 4 waiting line was a big no-no a few year ago. Now people were sleeping there every night and rangers didn't mind.
