1. Try the area around 7th Ave to 9th Ave along Irving - ie the Inner Sunset - to get a flavor for the area watch the N-Judah unload on Irving at 7th St, and grab lunch or coffee at one of the many shops.
2. Slightly east is Haight/Ashbury - well known area so i won't describe
3. Inner Richmond near Clement/Arguello - lots of bars/restaurants/coffee shops on both Clement and Geary
4. There is now a glut of rentals on the market in SOMA/South Beach as Sellers can't sell so they are renting instead. Anything from 4th Street eastward and the closer to the Embarcadero/King Street the better for restaurants/shops/bars.
5. Inner Mission - thriving night life along Valencia and Mission from 15th St or so to 25th St or so.
5a or 6 is the nearby Mission Dolores neighborhood - the closer to the Mission the more reasonable the rents
Honorable mentions.... Lower Pac Heights near Fillmore & Bush/Pine. Hayes Valley near Gough/Hayes. The western/flat part of Nob Hill along Polk near Bush/Pine. There are many other great neighborhoods, but most of them are either more expensive, or don't have as much to walk to. Whereas the above should have some of the most modest rentals and a LOT to do within easy walking.
For a little more in-detail advice, it all depends on what you like. If you're into bohemian, divey bars and hipstery nightlife, check out Lower Haight or the Mission. If you like a frattier, meat-markety scene, the Marina, Russian Hill or Pac Heights (around the Fillmore). And if you want to get a little off the beaten path with some more bang for your buck in terms of space, check out the Inner Richmond at 9th and Irving.
I'm a resident of Inner Sunset and I love it there. The nightlife isn't too crazy or anything, though we've got a few good bars in the area and there's always a good place to get a bite to eat and many restaurants stay open a bit later than other places in the city. It's also right on the N-line so you're always just a 20 minute ride from downtown. Parking has also improved recently as they've just reduced the street cleaning for many of the avenues from once a week to twice a month. One of my favorite aspects of the neighborhood is it's proximity to some of the best areas of Golden Gate Park.
The link below will give you the ultimate breakdown of the neighborhood and includes a list of all the restaurants, shops and bars. It does say that Sunset has some of the worst weather in the city, which is sort of true - on a foggy day it's always a bit foggier and colder there. I used to really mind that but it hasn't bothered me lately. Maybe I've just gotten used to it or maybe it's because we've had great weather this past week.
Since you mention modest income, I recommend Glen Park. It's just south of Noe Valley and the rents tend to be a little lower. Parking is also easier. Glen Park has great restaurants, coffee shops, & an organic grocery store to name a few features. There is also great access to freeways 280/101 & Glen Park BART.
The Sunnyside & Mission Terrace are adjacent to Glen Park that you may want to consider as well.
Craigslist is a great resource to check for rentals and you can search by neighborhoods. The neighborhoods mentioned by the Binnings Team are great but most of them are going to be at the very high end of the rental spectrum. You can get a sense of pricing through Craigslist.
Welcome to San Francisco!! It's a fantastic city!
Depending on your price range, there are some great deals to be had these days. Many sellers, disappointed that they aren't getting the prices they want for their condos, are deciding to rent them out instead, until the market comes back. So I suggest negotiating your rent, if you end up moving.
Try to visit the following areas in person (in no particular order):
Pacific Heights (around Fillmore)
Noe Valley (around 24th street)
I know it sounds like a lot, but but there are over 80 neighborhoods in SF... so this is just a handful that I recommend based on the info you gave.
Here is a map with the neighborhoods. Click in to see the areas. Good luck!