What is supposed to hurt when you run?

Strictly speaking, nothing should hurt *that* much - a little soreness is ok, but if your knees start hurting, or your hip flexors, or your feet, be careful.
Two tips that help me a lot:
1. Make sure you have good, high-quality running shoes. I strongly recommend going to a running store - not a generic sport store but an actual running store - and having them watch you run in different pairs of shoes to make sure you're in the best pair for you. There are stability shoes, motion-control shoes, and other kinds and depending on your feet and your stride they'll be able to tell you what's best.
2. Also very important is to make sure you're following a regular stretching routine. If you run without stretching you greatly increase your risk of getting injured. There are different philosophies out there but I believe that stretching after a run is best.

Good luck!

Well, running is one thing I DO know about. I have been running more on then off since elementry school. And was captain of my track team in high school (I'm not trying to brag, I'm just trying to give credentials), I was lucky in that I had a good coach who taught me not only how to run, but what to expect when I do run. So I'm more then happy to pass that info along.

As to aches and pains with running. It can happen in many places indicating different things. For you Caroline, it seems your calves are sore after you run. I would have been more concerned if it was your hami's, not a lot, but some. Soreness is to be expected when picking up running again. And from time to time, when I push myself, I still feel it a little. Soreness is generally never a problem in any of your major muscle groups after a good run. Basically if your calves are sore, one of two things is probably causing it. 1 is they may have (over the years) "bunched up". Without proper stretching muscles tend to bind up and become less elastic. This is common, it will take a while for them to lengthen again, but they will, just don't push it, if you start to cramp up, or if you feel any "pop" then stop immediately and massage the muscle (for cramps), or test the muscle by walking lightly (for a pop), if it hurts when walking, get off it and consider seeing a doctor as you may have torn something. 2 is they may just have atrophied somewhat. The soreness in this case is more just the muscle building up again. This should go away in a few weeks at most, if it's still bothering you after a month or so, then there is something else going on and you should consider seeing a physical therapist.

One thing to remember, and I see so many people at the gym on the tred mill that don't do this, is ALWAYS always always do a good stretch before you jog. stretch the hami's, the groin, the shins, the calves, and the quads before any run (stretching the back is a good idea too). here is a good site for stretching exersises.
http://www.coolrunning.com/engine/2/2_1/126.shtml

Oh, and it's a good idea to stretch after too. Not as big a deal for running, but vital for weight training. Do make sure you do a few minutes cool down though, it's important to give the muscles and your blood pressure a ramp back down to normal, if you just stop, your cooling off too fast and you can stiffen up and hurt yourself.