If you see some dandies growing somewhere that you know to be chemical and dog wiz-free, by all means, pick them. One thing to watch out for is the "false dandelions", Cats Ear and Hawks Beard. They usually show up later in the season, but you don't want to accidentally mix them up with the real thing. You can consult a field guide or just use google images to verify what you've got, but odds are if it's this early in Spring, it's a dandelion.
The only part that you need for the wine is the petals, but I usually pop the whole flower head off, then pluck the petals from the head while loafing on the couch, watching Netflix. It takes forever and a day to pick and pluck enough petals for a batch of wine, but I assure you that the end result is worth the trouble!
I'm trying to locate the recipe book that I used for last years batch - still looking. It's probably in storage somewhere still, but don't let that stop you from seeking out a recipe of your own elsewhere. Even if you can't find one you like right now, go ahead and pick your petals and freeze them, because the genuine dandelions aren't actually around for that long. You'll need quite a bit. If I recall correctly, my recipe called for 1 quart of petals (a spaghetti sauce jar full) per gallon of wine - and I made a three gallon batch!
Your neighbors will see you picking weeds on the side of the road and look at you funny (mine did anyway) and your fingers will be sticky and yellow for days after, but with relatively little effort, you can turn a weed into wine! It's the ultimate in upcycling! :)