Running obviously burns calories quicker than walking. However you're probably going to be able to run for less time than you can walk, so you may end up burning the same (or fewer) overall: better to walk for 30mins than run for only 10mins.
Running will, however, tone you up better (quicker), and the more muscle you have, the more calories you will burn at rest (i.e. the higher your metabolism will go). This means that in addition to cardio (walking/running/swimming etc) you should do some sort of weight training to boost your muscle mass (this is particularly important for women, whose fat mass naturally increases as they age due to hormonal changes). Free weights (can of beans, bottles of water etc) are great since they also work your balance, and you can also use your own body weight (the plank position, (modified) push ups) etc.
Running will also work your heart and lungs more than walking (assuming you do both for the same amount of time), and so is better in that event.
Running can, however, harm your joints, and so walking may be better if you're not used to running / will be exercising on harder surfaces / are older etc.
You could always try intervals, switching between walking and running.
If your feet hurt after walking, you probably need different shoes.
Heart rate wise, the simplest way is to work out your 'maximum heart rate' : 220 minus your age. You should then aim for anything from 50% (beginner) through to around 85% of this (experienced exerciser) for a consistent amount of time (30mins 5 times a week as a basic guideline). So if you're 20, your max heart rate is 200, so you should be exercising at between 100 and 170 beat per minute for this amount of time.
Having a low heart rate at rest is a good sign: it suggests a strong heart (the amount of blood your body needs doesn't change, so fewer pumps suggests more blood is being pumped each time). But if you have a low heart rate whilst exercising, you're not working hard enough... HTH