STYLETHREAD -- LET'S TALK SHOP!

It's insanely expensive and yuppified, but the Marina sounds like it fits all of your criteria. It's right off the Embarcadero, near the Presidio (great for dog walking), there's a Safeway right there, lots of dining out options, and public transit options (pretty much anywhere in the city is accessible by public transit, just depends on how long you want to spend on the bus/Muni/BART). I think the tough part will be finding a place that accepts dogs, although I do see tons of them walking around that area.

I lived across the street from GG park in the Inner Sunset. Perfect neighborhood, IMO. If you're going to have a car, it's the easiest neighbhorhood to find parking for free, and easy to get out of the city to the peninsula or Marin. It's also close-ish to the beach. And, it's affordable (by SF standards). Lots of restaurants, some good shops, one large local chain grocery store and lots of smaller Asian markets, and not too far from a Trader Joe's and a Safeway. The Sunset gets a bad rap for its weather, but it's really not worse than most of the rest of the city, in terms of fog, and it's much better than the Presidio. On the other side of the park (the north side) is the Inner Richmond, which somewhat mirrors the Inner Sunset. Over there are better restaurants and better shops and the best yoga studio in town, but the weather is a bit crappier, the people are a bit less friendly IMO, and the neighborhood to me feels a bit darker and dirtier.

If you want to live near a sizable park, your only options are GG Park and the Presidio. The other parks are pretty small.

Mass transit is pretty good throughout the city. In the Inner Sunset, you have various buses plus the N-Judah subway/tram (check out SF Muni's website for maps of bus and tram lines and BART's website for their service). To give you an idea of time - if you look at a map and locate the beginning of Golden Gate park - assume that anything West of the beginning of Golden Gate park to the beginning of 19th Ave/Park Presidio will take you 30 minutes (non-peak) to 45 minutes (peak) on public transport to get into the financial district/Soma (where Acad of Art is located). West of there will take you another 10-15 minutes. The neighborhoods north of the Financial District will take you

A brief rundown of my opinion of other neighborhoods, FWIW:

Noe Valley - Very dog friendly, lots of young families. Stroller valley, it's called.
Hayes Valley - Very hip, good restaurants, close to theater, but bordered by some slimy areas.
Marina - One big giant frat party, from the man wolves in the clubs to the girls who run on the weekend in full makeup.
Laurel Heights / Pac Heights - Really nice, great shopping, good restaurants, but quite pricey. also, weather can be pretty awful.
SOMA/Mission Bay - Easy living (garages, modern amenities - it's almost unheard of to find an apt. in SF with a dishwasher) but can be a bit expensive, and you don't get the SF charm. Dog patch area is pretty cool, but getting a bit far out, and can be seedy.
Mission - Very hip, best restaurants and nightlife in terms of art and music, but lots of drugs and seedy parts
Dolores Park - The Mission, but slightly safer and cleaner.
Castro - The "gay" neighborhood. Really nice. Lots of dog-friendly stuff and feels very safe. Some good restaurants.
North Beach - Italian neighbhorhood. Very quaint, good restaurants, but pretty packed. I love it, but it would probably feel claustrophobic without nearby parks.
Cow Hollow - Cute, good restaurants and shops. Not very affordable. Borders the Marina.

Avoid the western addition / Japantown area. There are some nice pockets, but mostly it is not nice, and is regularly reported as SF's most dangerous neighborhood.

Obviously there are more, but I've already gone on waaaaaaaaay too long. HTH

I agree with most of Miss Mabel's exhaustive list.  I wanted just to note that North Beach does have one really big park--Washington Square Park, right in the center of the neighborhood.  It does get pretty packed on the weekends, though, because it is truly the "center" of the neighborhood and everyone goes to sit out there.  Ditto for Dolores Park. 

Noe Valley is really nice and walkable, and would be perfect if you are into parks and having a dog-friendly environment. 

All three of those neighborhoods are very accessible by public transit.  One thing to note if you are going to look in North Beach, it's pretty commonly known to be the absolute worst neighborhood for parking in the city.  If you rent there, you might want to make sure you are geting a garage with your apartment, because it will be next to impossible to park on the street.   It's really a charming, quaint  "old SF" feel, though--I love it there. 

Good luck with your move!