You should be able to get at least two cuttings on most perennial herbs, even in northern Minnesota. Cut them leaving about 1/4 to 1/3 of the stem on the plant... there should be at least a couple pairs of leaves left (not a problem on something like thyme, generally, but for large leaved plants it gives you something to aim for)
As far as what parts to use, generally the leaves are what is used for cooking, but if you are using fresh herbs and the stems are tender (basil being one which often qualifies) you can add entire sprigs to the dish without a problem.
You can give the plants a boost by adding some compost or other *mild* fertilizer when you cut them this time, but don't fertilize them when you take a second cutting later in the summer... you don't want to encourage new, tender growth close to frost.
Generally, I dry entire branches of the plant, and then strip off the leaves later in one (messy!) operation before putting them in glass jars for storage. Don't crush the leaves or worse, grind them, before storage. Store them as close to whole as possible, and grind or crush them just before use.. they'll stay much fresher and more potent that way.
For annual herbs (basil, and something like rosemary if you aren't going to try and overwinter it indoors) you can harvest early and often, because you aren't worrying about trying to keep the plant's root reserves healthy. With basil, you need to keep clipping it to keep it from flowering, because that stops leaf production once it starts to set seed.
Have fun!
Summerthyme