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Based on the type of riding you describe a hybrid bike would be a good choice. Flat bars in a more upright position with some semi-knobbie tires (for the grass and dirt times) would be good. Many hybrids come in around your budget or below. Look at Bikesdirect.com for some good deals. Look at the components and compare several. Test ride several bikes to see how they fit and perform.
To go with the bikes you will need helmets and some other clothing too.