The best option, if it's available to you, is to use sod from another (inconspicuous) area of your yard. For example, if you have an area behind the garage that no one really sees, you could remove the sod from there and put it in the area where you need grass. If you need to mix and match from a couple of different areas, that's okay too. The area you dig the grass up from will probably eventually get covered in grass (or a grass/weed mixture if that's what your lawn consists of), so it won't be a big patch of dirt forever--although the recovering could take a few years. But if we're talking about an inconspicuous area you don't care about much, then it probably won't matter to you. (Or you may decide that temporarily sacrificing one area might be a good trade off to improving a more visible area).
If you have an extremely neat yard and are looking for perfection, you'd wanna go with grass seed, of course (on which I don't have much advice). But if you're looking for a faster, easier, cheap, low-maintenance, generally reliable solution, a sod transplant is the way to go. Growing grass from seed is a hit and miss process. Whether you have success will depend on many factors--choosing the right seed mixture for your soil type and sunlight, watering regularly and properly, and probably a lotta luck. Even if you do everything right, you could end up with it all dying, or having it grow very sparsely. Of course the same thing can happen with sod, but IMO, sod transplants are easier for beginners.
If you get sod from your own yard, you'll know that your new area will match the rest of your lawn in both color and growth rate. And you'll know that it's the type of grass that does well in your yard. Depending on the specifics of your lawn, you probably will have to treat it with kid gloves most of this year, not walking on it, watering it well if you don't get enough rain, and not mowing it until it has rooted in, which will make it look unattractive for a while. But this will be the case even if you plant from seed--no matter what, that patch is going to have to be a "work in progress" for this year.
I've got a lot of property in the country, and when I moved in it was a mess. I've had to patch up countless grassless areas over the years, and even though I have a green thumb with other plants, I have trouble with grass seed. That could be due to the type of soil, sunlight, and/or climate I have, and maybe other people have a lot of success with seed, but my success rate with sod transplanting is better than 90%.
The best time to do the work--whether seed or sod--depends on your climate. I'm in the midwest US, and I prefer to do it in May or June. If you transplant, you'll want the area-to-be-covered to be dug down (or built up) to a depth that, when you put the new pieces in, the pieces will be just slightly above the level of the surrounding grass. This depth will be determined by the depth of the plugs of sod you use. Use a spade to cut around a manageable-sized area of the grass to be moved. Sod plugs are heavy and can break apart if the plug is too large, so I usually cut it in pieces about 1 foot by 1 foot. I dig down deep enough to get the majority of the root system, sliding the spade under the plug to release it. (I have different types of soil and different types of grass on my property, and sometimes I get nice neat little plugs that are only 4-5 inches deep, sometimes they are 8 inches.) Then I plop a plug into the new area and take a gander at what level it's lying at. If it's too low or too high, I take the plug out and build up or dig down till the level is right. You really don't have to be exact, close enough is good enough. Just shove a bunch of plugs into the spot to be covered, tamp them down with your foot till they're a little above the rest of the ground level (to account for settling), and you're done. In the areas I've done this, even though the area looks choppy and uneven the first year, and sometimes even gets a bit yellow and looks like it might die, the next year I have an brand new area of grass that is indistinguishable from the yard around it.
