Although it is absolutely possible to arrange your own visit to Taiwan, however, in most cases a better option is to go on a package tour, or to get a travel agency to plan and book the trip for you.
There are many travel agencies and tour operators operating in Taiwan, all at different price levels accommodating your vacation plans and catering for different budgets and tastes.
Once you know how much you want to spend, where you want to go and what you want to see, you should start searching on the internet to find some of the possibilities available. Also it would be a good idea to visit travel agents to learn more about alternatives and get some good advice. Maybe the agent can suggest something better than you have already found, and if not, you will know that your own search has already covered your wishes.