I've recently started writing and I told my family. They said that they were proud and that they would support me but when they weren't paying attention I overheard them saying they thought I wasn't going to stick with this and that I'll soon go back to my school work and focus on getting into college. I don't want to spoil the storyline or any of the plots I've thought of, but I want to tell them that I'm really doing this and make sure they believe me. How can I do that and still not spoil the whole story to them?
As you may well have realized already, turning a premise and a plot arc into a fully-realized story is a long and arduous process; it's not uncommon to lose heart partway, so your family's skepticism is, however uncomfortable, an understandable reaction.
The best cure for skepticism is results. Fortunately, you already have their promise to support you, and this is a good thing; family almost always makes for biased reviewers, so you'll eventually want to ask others, but almost any input is useful when you're just starting out. You don't need to discuss the whole story with them (although if you have one person you trust to give you honest feedback, I would recommend doing so), but try to get them involved in the process.
Write the opening chapter, polish it until you have something you're happy with, and ask them for their thoughts. Does it pique their interest? Do they want to know more? Is the protagonist and/or viewpoint character intriguing or sympathetic? This accomplishes three things: proves you're actually writing and not just thinking about it, gets you potentially useful feedback, and shows them that you're taking the process seriously enough to do your research. Then take what you learned and work through the next few chapters before asking for more feedback.
As a bonus, this kind of schedule can also help you motivate yourself, knowing that people are interested in your work and expecting more of it. (Assuming they are, that is, and if they're not, you have the opportunity to ask why and work on that early on. Keep in mind differences in taste, though; ask friends with similar tastes if you need to.)
Ultimately, all artists need to have faith in themselves and their work, and not rely on external validation, even from friends and family members. Writing is a difficult and demanding art, and most writers, even well-known ones, do have second careers to help make ends meet. Writing a best-seller overnight is a one-in-a-million shot at best (most successful writers work for years before their first sales), so your family is not wrong to want you to have other options, even if they have full faith in your abilities. It's quite likely that you might be "working in the trenches" for a long time, even if you eventually become a big success. I say this as someone who was completely convinced I would be that one-in-a-million success at your age, yet who is still working towards his big break twenty years later.
It's also the case that there's a real difference between having good ideas for stories and plots and being a writer. Many people have good ideas, but the hard work is turning those ideas into successful stories and books. So keep in mind that whether or not you share your plots with your family, you won't really have proven your point until you have published work to show them.
Finally, college is a good place to improve as a writer --most schools offer some form of creative writing as a major or a minor. So you can pursue your dreams and address your parents' fears at the same time. None of this is to discourage you, or to suggest that you can't or won't be a great writer. But you will never get far as a writer if you're dependent on the belief, approval or praise of those around you. You have to find motivation within yourself.
One way you can show them you are serious is by actually writing, even if you don't show them anything for a while. Give yourself a goal, whether it's number of words or X amount of hours, that you want to do every week. And then do it. And let them know you are doing it. This doesn't yet have to do with whether you are good (in your mind or anyone else's) but it shows that you are serious about it.
Also, as @ChrisSunami mentions, college is a good place to work on being a writer. Many ages ago, my undergraduate work was at a liberal arts school where I could focus on writing. I learned a lot and met a lot of great people. I got to talk to people with actual experience, which can be hard, especially when you are young. Unless your family is pushing you toward studying a specific subject in college, consider combining your writing and their desire for you to go to school.
In reality you can't with words (ironically) you just need to do it, keep your writing time for you. Don't run off and do a chore or if they need to pull you aside for X unless its like an emergency or a grave thing tell them, "No, I'm writing."
It took me over 2 years of writing and not dropping everything to tend to X for people to not only get it but then want to know more and actually listen if I chose to give them a few minor details here and there as my story is still evolving for me so I too don't want to tell them major things that might change after the draft is done as this is my second one and a major rewrite so I know change is certain but I'm not yet sure where and how.
You need to not let their comments get to you actions are louder then words here their support either will or won't come in time and you can't make them support you beyond superficial means. You believing in you is required and will always be so and what you need most. If you want to write and publish, work on your draft, get your story after that nailed down, and then keep going to fine tune it, you have a long road ahead worrying about their approval will not do you good on that.