Concerning biking: is 10 miles too far for a beginner? (Read the details, then answer, for goodness sakes!!!!)

I recently got a really good ($300) Trek bike, with suspension on the seat, front AND rear handlebar brakes, room for lots of additions, and plenty more. Then, just today, I got a computer unit for the bike so I can see how fast I'm going. So now I have some really good motivation to start a biking excercise regimen. Later in the afternoon, I rode a little over 5 miles and was out of breath about twice, and even ended up stopping at a crappy water fountain along the bike trails that outline the Big Sioux River and its prospective diversion channel in the small metro area of Sioux Falls, SD (my hometown and current city of residence, population 145,000, metro pop. 230,000). Is 10 miles on that trail going too far for me? And if I decided to eventually try and ride the entire 20 mile loop without stopping, how long into my regimen should I wait before doing so? Please give me reasonable, thought out answers that make sense, not something like, "I don't think it's the best idea for you"!!