Where is the best neighborhood to stay in San Francisco?

I suggest staying downtown around the  Union Square area.  From there, public transportation to most parts of the city is pretty easy.  You can take the BART subway train, MUNI bus, and it's very easy to hail a cab if you need to.  It's pretty much the center hub for public transportation.  Just don't plan on renting a car there because parking is a nightmare!  That's close to a lot of shopping at  Powell Street, San Francisco Chinatown, and you can also walk to the  Ferry Building, which is a very famous spot in San Francisco full of specialty food stores.

Other neighborhoods in that same area that you can look into are  Nob Hill,  Financial District, and  SOMA.  With SOMA, look for places that are closer to Union Square/Market St. because otherwise, SOMA also reaches out to the water away from public transportation and attractions.

If it's your very first time in San Francisco, the  Fisherman's Wharf area could also be a fun place to stay, but it's mostly very touristy and a little more difficult to get around than the downtown area.

One neighborhood that I would try to avoid is the  Tenderloin.  Though very close to Downtown, things get sketchy really fast there.  There are some okay parts there, and it's getting better, but there are also some really scary parts, too.  I'm okay walking alone there in the day time because I'm familiar with the area, but I wouldn't suggest anyone new to the area walk around there by themselves or even in a group because you don't want to make a wrong turn there.  It's a very small part of San Francisco, though, so anywhere else, you'll fine :)

San Francisco's great and you'll have a lot of fun.  Let me know if you have any other questions! 

ETA: Some of my favorite places in SF since you asked in a comment, and I swear you could even get a glimpse of all of these in a long weekend, so 5 days is perfect to really enjoy everything!

• Ferry Building: Try to come here on your Saturday morning in SF because that's when the famous farmer's market is!  It's not just for buying fresh, local produce.  There are also a ton of great gourmet food stands (a lot of which have actual restaurants), so you can experience the awesomeness of San Francisco food.  Inside the Ferry Building itself is also great to experience.  If you like cheese, got to  Cowgirl Creamery And if you like coffee, you have to go to  Blue Bottle Coffee
  
• Cupid's Span: While at the Ferry Building, walk over to Cupid's Span for a great photo op of the sculpture and the Bay Bridge.
• Waterbar: If you like oysters at all, walk a little further down to Waterbar.  It's a gorgeous restaurant with a beautiful view of the Bay Bridge and the water.  It's a pricy restaurant, but they actually have an awesome happy hour menu.  From 11:30 AM to 5:30 PM every day, they have $1 oysters!  Try to go earlier in the day though, because they don't take reservations and they get packed.
• Palomino Restaurant: If Waterbar is packed, go across the street to Palomino.  Around lunch time is when the sun hits their patio the most, so try to go there then.  Another gorgeous restaurant that looks pricy, but actually has an awesome all-day happy hour menu.
• Justin Herman Plaza: If you walk across from the Ferry Building to the right, you'll see the Justin Herman Plaza.  The egg head sculptures are fun to photograph and there's an amazing industrial-looking waterfall structure that you can actually climb and also walk through (just discovered this the other weekend!).
• Fisherman's Wharf: There's the  Pier 39 here which is full of restaurants.  Tons of seafood here and you should at least try clam chowder from a bread bowl, which you can get at most restaurants there, or even from a street vendor. You can see sea lions here.  There's also an  In-N-Out Burger there if you like burgers.  It's a well-loved chain of burger restaurants that's mostly in California.
  
• Alcatraz: You can take a ferry over to this old prison island.  It's pretty fascinating.
  
• Golden Gate Bridge: Something else that's fun to do is to rent a bike and bike across the Golden Gate Bridge.  You can also walk across it.  It's typically an all-day thing, though, if you can allot the time for it.
• Golden Gate Park: There are tons to photograph here because it's a huge park!  De Young Museum is beautiful on the outside even if you decide not to go in.  Stow Lake is gorgeous, as are the  Japanese Tea Garden and the  San Francisco Botanical Garden.  The outside of the botanical garden is beautiful, and if you want to go inside, it's only $7.
• If you want to go barhopping, the  Mission District and  Polk Street are great streets to walk through.
• For a fun place to picnic, you have to go to  Dolores Park midday to picnic!  It's the thing to do!  Grab a bottle of wine  Bi-Rite Creamery, maybe some ice cream at the  Bi-Rite Market, and a sandwich from  Ike's Place or a burrito from  El Farolito and just laze in the sun.

Nob Hill,  Noe Valley, Lower Pacific Heights, Cow Hollow, and the Marina District are all great neighborhoods.  The perk of Nob Hill (snob hill is the nick name) is that it is close to Union Square and all the shopping and restaurants there.  As for the Marina District, there is cute boutiques and fun bars as well close to North Beach, Golden Gate Bridge and Crissy Field.   Noe Valley is an adorable nice neighborhood that has cute restaurants and more of a family vibe.  You can't go wrong with any of these spots.  Hope that I have been helpful :)  Have a nice trip!

Lower Pacific Heights is a nice place to live with the busy Fillmore Street which is good for nightlife. Few parts of Pacific Heights  are upscale. SOMA SoMa is where all the techies are. Nob Hill is also a decent place to live considering the public transportation. But for May 2014, please start looking for places from Mar or April. Finding an apartment in SF is crazy. It took me 1.5 months and 30+ visits to finally get one. There are shady ones, there are good ones with 10+ people lining up before you. 

I lived on  Nob Hill for a time -- I highly recommend The Stanford Court San Francisco there. The views from the hill are great -- and if you're lucky, maybe one day you'll get to watch the fog roll in and swallow the city while you're high up on the hill watching. It's beautiful. 

Union Square is also full of activity (shopping mainly) and a variety of hotels. Although, if I'm not staying on Nob Hill, I'd stay in  North Beach close to  Fisherman's Wharf.

If you can, ditch having a car --- parking is a pain and it's not uncommon to have it broken into. Public transport is pretty good within the city -- and taxis are plentiful (as it's a small area, they aren't even that expensive). 

Hi. I like  North Beach it's a nice trendy area close to downtown. Walking distance to Fisherman's Wharf area and it has lots of amazing restaurants and bars. Little Italy is a nice area to eat and Washington Square Park is a nice place to hang out in around North Beach. There is a lot of public transit near. I would recommend hiring a bike and biking around the city and over the Golden Gate Bridge. 

For public transport, have a look at http://www.sfmta.com/getting-around/transit

I would definitely consider the Union Square area.  Most of the public transportation is there, including the famous cable cars.

Hayes Valley was a perfect place for us when we were there. Close to public transportation, felt like a neighborhood, lots if great stuff close by. I highly suggest it.

Union Square area has proven to be my best location......anything you might need just a short walk.  Bart is right there to take you to the outskirts, the Cable cars can get you down to Fishermans Wharf, etc. etc.

We enjoyed staying in Dogpatch. Easy access to everything via Muni, bus, or walking. Great neighborhood, close to  Portero Hill where there are several great restaurants and little shops. Dogpatch also has some great spots for eating and browsing.

This is very subjective so I will do my best to describe your options: 

- Hayes Valley: Awesomely centrally located and on the better end of safety, issues are there are few hotels/places to stay. Good for couch surfing 

- Tenderloin: Terrible place to stay in regards to safety, but the low(wer) rents allow for a few hostels, which are few in SF. During the day it's safe and its central, though at night I recommend a taxi to the hostel 

- Union Square: Most hotels are here, and it's central and a bit touristy but really great to get around the city. If I could afford it and didn't want the 'local' experience, I'd stay here 

- Mission District: Mission is a mix of different cultures, but is predominantly driven by Central/South American culture mixed with hipsters. You can find a cheap burrito and an overpriced coffee, depending on where you are. I love this place for the Central/South American culture and highly recommend trying smaller restaurants that cater to the locals that have lived there for awhile. Not many hotels, but things to find.  

- SOMA: More expensive hotels close to market street, these are safe and have everything you need. They are central and convenient to the transit. Be careful with Soma, the farther from market (and some areas by 8th street) can be a bit dodgy. Totally fine, but one must be aware there 

Places I don't recommend: 

- Fisherman's Wharf: no locals actually go there, and they are just trying to get money from tourists 

- Marina District: Though beautiful and has a few motels/hotels, they are far from central SF 

Hope this helps - message me if you need anything else. Also for those that disagree with my statement, I always appreciate feedback/a different point of view so please let me know/comment otherwise!

My favorite SF neighborhood is Hayes Valley. It's near Civic Center so lots of transportation options, and very un-touristy. It's flat, walkable and has lots of independent restaurants and shops. Another area I like is Parnassus Avenue, near Golden Gate Park. Don't go to Fisherman's Wharf, which is a total tourist trap. Union Square has every retail shop you can visit in any other city in America, so why go there? The Mission District has some local flavor, good shops and restaurants, but I find it still a little rough. Dogpatch is really fun in my opinion, it is in transition. It's all a matter of taste. SF is a small city with excellent transportation options so you can easily explore it. Have fun.