There're  over 300 wineries in Sonoma County, so one way to narrow it down is to pick a varietal that you love or just pick a geographical region near something else you want to see each day--figure on 4 (5 max) wineries a day, much more will just overwhelm you.  

So for example, if you like zins, Dry Creek Rd(outside Healdsburg) is awesome. The valley itself is about 1 mile long, narrow, and full of small wineries. If you like cabs, Sonoma or Alexander Valley. If you like pinot noir or chardonnay, try the Carneros region (south end of the Sonoma Valley) or Russian River--there are a dozen or so wineries up Olivet Rd, you can combine an Olivet road tasting  with a hike in the redwoods: at River Road turn left and visit Korbel Winery (nice gardens, free tours, great deli) and continue on to Armstrong Redwoods.

All of these areas will offer other varietals, as well, of course, but each has microclimates that work well for a selection of varietals.  

If the weather is nice take a picnic. Most wineries have lovely picnic grounds (the only requirement is if you have wine with your picnic it has to be the wineries--it's not to pressure you to buy; if you have outside wine or beer , it's a liquor law violation for them.) If you plan to picnic in dry Creek valley, pick up your picnic in Healdsburg or where ever you're staying, there's only one grocery store/deli in Dry Creek Valley itself, and it;s usually mobbed. 

You can also narrow it down by kid or pet friendly wineries, historical wineries, etc. 

Post again f you'd like more specific recommendations for a narrower focus. 

Last bits of advice: some of the large wineries offer free tours, otherwise stick with the smaller ones. You'll get more attention, and you'll discover wines you can't get at Costco. And check local events listings or the wineries websites; they may have events while you're here that can be fun (live music, etc.) 

You have many, many to choose from.  It depends on whether you are in for a big tour or just want to taste some wine.  One tour is enough for two days.  Jordan Vineyard & Winery  has $30 to $120 tours and tastings, but both the tours and Cabs and Chardonnays are very good.  We stayed a couple nights in Healdsburg summer of 2013 and enjoyed tasting at Quivira Vineyards & Winery excellent Sauvignon Blanc and Rose and nice winery and garden, Preston Family Vineyards , Bella Vineyards & Wine Caves good Zin and caves to visit, and Clos Du Bois Winery a big winery with wines of all prices and some interesting ones only available there and good people.  Healdsburg is a great town and I really like Scopa for casual Italian food, but make sure you make a reservation.

I would highly recommend Cline Cellars and Jacuzzi Family Vineyards in Sonoma. They are right across the street from each other and are both beautiful. The Cline property is more of a home grown and peaceful farm vibes. Jacuzzi across the street feels like you've walked into a square in Italy. Both have incredible wines.

We adored Gundlach-Bundshu Winery, the setting is beautiful as well.  Definitely plan to bring lunch with you and enjoy the scenery.  It isn't right in Sonoma, but it is on the way to Napa-Domaine Carneros.  It sits up on a hill overlooking the valley, has a lovely gift shop and a huge outdoor patio.  Nice way to end the day-with bubbles!  I've always been a fan of Deloach Vineyards-their estate stuff is really pretty good, and can be found in most places once you leave-always nice to recognize wines later that you are familiar with.  Seghesio Family Vineyards makes some of my favorite zins.  Hook & Ladder Vineyards & Winery is connect to the DeLoach winery (I believe when Deloach got too large, he started this one).  Excellent reds, great blends.   Buena Vista Winery produces some really solid juice-good Chardonnay (my brother belongs to their wine club).  Frei Bros Winery Rd makes a killer barrel-fermented charonnay!  Joseph Swan Vineyards has some excellent high end stuff.  Schug Carneros Estate Winery makes good stuff for mid-range pricing.  And Pride Mountain Vineyards is amazing!!  Their merlot and cab are some of my favorites!

On my last trip to Sonoma wine country, which was also my first, we wound up at the following three wineries which I enjoyed immensely:

1) Ridge Vineyards - Lytton Springs

2) Papapietro Perry Winery

3) Joseph Swan Vineyards

We also visited Wilson Winery, which was just way too busy and crowded to be able to enjoy the experience that day.

Also keep in mind that Russian River Brewing Company is also nearby in Santa Rosa.