Let me get the business out of the way first: what schools can you afford? have your parents given you a firm figure they can provide you each year? Have that talk first because you surely don’t want to visit a school you cannot afford to attend. After you get that firm figure, keep in mind that VERY few American colleges provide ANY aid to international students. If you need aid, you will find it very difficult to go to school here unless your scores and extracurricular were astronomical (and they’re not). So the money talk may settle some issues for you, but here are some places where you will find lists of American schools that give SOME aid to internationals (generally, the most prestigious schools provide the most aid):
http://www.educationusa.info/financial-aid.php
http://www.desperateguide.com/us/top-25-financial-aid-colleges-in-us-for-international-students-need-aware
http://www.usnews.com/education/best-colleges/the-short-list-college/articles/2013/09/19/colleges-that-give-international-students-the-most-financial-aid
Not to the question of admission: You’ve nice scores, grades, and extracurriculars. Nothing out of this world, however, so I would caution you against expecting to go to a Top Twenty school, as many internationals (and Americans!) do. No one should expect to get into these schools.
Unless you have a lot of money, I wouldn’t expect you to need to travel to the US to check out schools. If you do come, you might consider doing that only after you’re been accepted to two or more schools you really want to go to and know what aid they will provide you.
However, if you have all the money in the world for travel, I would suggest you decide on visiting two Top Twenties, three or four match schools, and a safety or two (if you MUST go to school in the US). Those visits should be split among one ivy (Cornell might accept you more easily than the other ivies); one near-ivy LAC like Amherst, Pomona, Bowdoin, or Swarthmore; one match LAC like Haverford or Vassar; one flagship like UVA, Michigan, or Berkeley/UCLA (they’ll have a lot in common so seeing one is seeing many of them); and one midsize uni like Tufts, Brandeis, WashU, Chicago, or Duke (similar to seeing a flagship but not quite the same). Of course all of these suggestions could change depending on your preferences for size, region, major, etc.
You can use the SuperMatch feature of this website; look for it in the column on the left to get you started on building a list of interests. Then come back after you’ve done some research. One benefit of this process, you’ll learn that the US is a really big country with long jet flights between colleges (altho it pales in size to some others). What you’re proposing is very expensive.