You have to consider a couple of factors when choosing which
variety of grapes to grow. Grapes for the northern US are
self-pollinating but some Muscadine grapes (southern US)
require planting a pollinizer. Most are subject to various
fungus diseases although some are more resistant than others
and some are altogether immune.
In Chicago there are many yellow jacket wasps in the fall and they love grapes so it is advisable to get a variety that ripens early before the wasp population explodes. To trap wasps I've tried something called The Yellow Jacket Trap by SureFire products. Their bait did not attract wasps away from the grapes however if you take a grape and force it through each of the four holes the juice on the entry holes will attract wasps inside, once inside they pretty much can't get out and they die if left there long enough. If you like to chase wasps around with a sprayer then put some dishwashing liquid in water and spray the wasps, if you get enough on them they will die in a few minutes.
To protect the grapes against birds of course you can use a large piece of netting but birds may find a way inside and even get caught in the netting. An alternative is to save the netted grapefruit/onion/potato bags from the grocery store and use a twist-tie to hold them on over the bunch of grapes. Birds (I guess) managed to get through the bags and pull them off the bunch, I suppose they got a few through the netting but most of them landed at the bottom of the bag. I've gotten the best results by using the netting and the bags.
If you need a trellis to support the vines you can build one with 10 foot sections of PVC/CPVC pipe that you can get from a hardware store (cheap). Drill holes in the pipe and use machine screws to hold the pieces together. On the vertical sections drill holes every 6 inches or so so you can string copper wire across the trellis to increase the support for the vines.
Another issue is when do you pick them? Jung has a tip that you should test the bottom grape of a bunch and when it is sweet enough, pick the bunch. The trouble is that green grapes are often unevenly distributed throughout the entire bunch, rather than just being on the bottom. I end up picking bunches early so as to avoid losing grapes to the ants, wasps and other garden monsters.) Reliance seedless grapes that are freshly picked can have a sharp taste but if you let them sit on a sunny window sill for a couple of days they will mellow. Even grapes that are only slightly pinkish will still be nice and sweet but they will lack the robust flavor of a really ripe grape. My Reliance seedless are ripening more or less around mid-August.
Reliance seedless will have a few blooms in July and August and these often mature before frost so you can have a few extra bunches late in the season.
Pruning is necessary to divert energy from the roots into the fruit. If you don't prune then the energy from the root will go out into lots of buds to form lots of leaves and the flower bunches that come out will simply fall off for lack of nourishment. With pruning the energy can "back up" into the flower bunches and they won't fall off. Pruning is done in late February or early March. After doing the cuts the vines may leak some water for several days to a week but eventually this will stop and it doesn't do the plant any harm. There are lots of rules for pruning. I'm kind of hap-hazard about it. Here is one plan taken from the 1959 Encyclopedia of Organic Gardening:
A general rule to follow in pruning is leave two buds the year the planting is done. Second year, prune off all growth except one strong cane and leave three to five buds. Tie this cane to the first wire of the trellis. Third year, leave two fruiting canes of six to eight buds each. Fourth year, leave two longer fruiting canes of eight to 12 buds each. Fifth year, three fruiting canes of 10 to 12 buds on each.
If you want to make more grape vines you can take cuttings from your plant and do as follows. Note though that modern varieties are patented and you're not allowed to clone this way although you can of course work from seeds. Take cuttings that are about a foot long and put them in a bottle of water so that about half the cutting is in water and bring them inside and put them in a sunny cool window where the water will not overheat. After several weeks the buds will open and the leaves will come out and after several more weeks roots will grow. When they have enough roots plant them in a small pot.
The fertilizer recommendations I've seen vary. One reference says a general purpose fertilizer will do. An old Encyclopedia of Organic Gardening says that potassium is especially important and you should apply 2 to 3 pounds of finely ground granite rock to the soil around a vine in the fall. Roots go out 3 to 6 feet from the base so spread out the rock. But where the heck are you going to find finely ground granite rock? To get potassium into the plant I've been using spray on kelp rated at 1-0-4. They say nitrogen is important too but phosphorus is not.
Grapes often get mildew (white powder) on the leaves or suffer from other fungus diseases. Problems like these can be minimized by growing the grapes in a very sunny area with lots of air circulation. Then there are varieties that are immune to these diseases and others that are only resistant, choose your variety carefully especially if you want to avoid spraying. If you have to spray the standard spray to use is Bordeaux spray, a mixture of copper sulfate and lime. These chemicals are safe in the sense that VERY small amounts of them inside your body will probably not do any kind of obvious damage. (Copper comes in Certs for instance, sulfate comes in glucosamine sulfate (said to cure osteo-arthritis), calcium (from lime) is just plain good for building strong bones.) On the other hand, copper sulfate is an acid salt, a solution of it is pretty much like sulfuric acid, the acid found in a car battery and it can cause burns. So you must be careful not to get any of it in your eyes, lungs, mouth, into cuts, into children or pets, etc., etc.. More than likely it would cause metal to corrode as well. The spray can be applied dry or dissolved in water.
I've also tried Soap Shield, a copper based soap from Gardens Alive. Its a liquid that comes in a 16 ounce bottle, normally use one ounce per gallon of water, spray once or week or after rain. I found this was effective in stopping mold that was destroying the berries. Its also easier to use, I think, than the Bordeaux spray powder. It decays away into copper plus fatty acids, both of which can be used by plants. I originally got Soap Shield to combat black spot on roses where it also seems to be pretty effective so you can actually put it to good use on other plants as well.
One source for grapes is Miller Nurseries in New York, they have quite a number of varieties available. I have never ordered from them, I got my Reliance seedless vine from Jung in Wisconsin. I suggest you browse a number of catalogs to learn all you can about each variety before making a choice, quite often one catalog will include details not found in another catalog.