Pickling vegetables is among the easiest ways to preserve your garden's (or farmer's market's!) excess. It's also one of the easiest things that beginning canners can make - just a handful of ingredients and about an hour's worth of time result in several pints of colorful, flavorful pickles. Plus, the basic recipe is extremely versatile. Once you cover the basic acidity and salt requirements you can add garlic or hot peppers, more or less dill (or some other herb - tarragon can be nice) or even make veggie blends. You can use whole, trimmed vegetables or slices or even shredded (especially nice for cabbage or carrots).
I use a simple recipe put out by the USDA (click to find copies of their Complete Guide to Home Canning). The USDA is responsible for advising home canners on the types of foods that can be safely canned, processing methods and techniques. Recently, the USDA dramatically increased the processing time for tomatoes to take into consideration modern low-acid varieties. So these are good people to know because they figure all this stuff out for you.
Another good source of information is the Alltrista consumer site. Alltrista manufactures a variety of canning equipment, accessories and mixes and they have a nice illustrated primer on the basics,as well as a good FAQ. In my opinion, you don't need much of the equipment or mixes they sell to be a good, safe, productive canner but you do need some of it and you should definitely take advantage of the benefit of their expertise.
One last thing before we get to the recipe: I've said it before and I'll say it again. Yes, I know your grandmother put up hundred of quarts a year using nothing but mayonaise jars and a hunk of parafin. Follow the latest rules anyway. Knowledge about food borne pathogens increases constantly and we know a lot more now than we did when your Sainted Grandma road her kitchen range. Another way to look at it: you'll likely never be sorry that you followed the modern guidelines, but you just might be sorry that you didn't.
Now for the good part - the recipe. First, fill your canning kettle to a level that would be 2 inches ABOVE the height of your canning jars when they're placed in. Follow the directions for preparing your jars and lids from either the Alltrista or USDA websites or that came with your jars (every new box comes with a set of basic instructions). Once your jars and lids are sterilizing, get to work on your veggies.
For pickled green beans, trim the ends and cut to a uniform length to fit into your jars (likewise for asparagus). I like to do peeled and trimmed carrots in coins (a food processor makes quick work of carrot coins). Brussels sprouts need only a quick trim on the bottom and onions should be peeled and trimmed. I've never done jicama or beets, although I'd love to, but they should be trimmed and cut as desired. I've also heard of people dilling broccoli, but I've never seen it for myself!
Pack your veggies into the hot, sterilized jars, leaving about an inch of "headspace" between the top of the veggies and the top of the jars. Tuck in a head or a nice healthy sprig of dill and include a clove of peeled garlic. A jalapeño pepper makes a nice, spicy touch - use red for green veggies and a green for carrots or beets. Some people also like to use a half teaspoon or so of dried pepper flakes, which makes for an interesting presentation and pretty spicy pickles.
To make the brine, combine in a medium saucepan: Two cups of water and two cups of white vinegar (5% acidity - regular supermarket white vinegar. In other words, save your fancy pants vinegars for something else) with a half cup of either pickling or kosher salt. Heat to boiling and fill your jars to a half an inch headspace. Clean off the rim of the jar using a clean towel dipped in hot water and seal using the two-part lids that the equipment directions have you already sterlizing.
Process in a boiling hot water bath (that is, put the jars on the rack only after the bath is boiling and then start counting your processing time after the water resumes boiling).
Once your processing time is completed, remove the jars (I use a jar lifter for this, but you can - carefully - use glove-type potholders or thick towels) and place on a tea towel to cool. You'll start to hear the little "pings" that indicate a good seal within 15 to 20 minutes. All of your jars should seal within 24 hours. Those that haven't should be refrigerated and used first. Try to let your pickles sit for six to eight weeks before opening for the best flavor.
So...what are you going to make? Do let us know how it turns out! I, for one, am on pins and needles.