Question: I got rejected from the schools I really wanted to go to. Is it true that it's easier to get into a school by transferring? I'm looking at schools like Harvard, Columbia, & Barnard.
Harvard and Columbia are almost impossible for transfers. These schools admit very few and tend to favor applicants with non-traditional backgrounds (e.g.., older students, disadvantaged candidates who first attended community colleges, etc.). Students who were just denied this spring as high school students have very little chance. The forecast is a bit better at Barnard but, even so, you will have to do extremely well in college--not just in the classroom but by distinguishing yourself in other ways, too (e.g., campus leadership, research assistantship, internship). This can be tricky for first-year students who are just starting their college careers and haven't yet had time to make a mark. Also, because sophomore transfers are admitted before their freshman year is even over, admission officials will only have one semester of grades to evaluate, which means that it's critical to get off to a running start.
Transferring is also, to a great extent, a "numbers game." This means that some years, if attrition is unusually low, there will be hardly any spots open for transfers. Conversely, when attrition goes up, so does the transfer figure. But at the most selective colleges, those open slots are almost always few and far between.
You have probably heard those "it's easier to get in as a transfer" rumors because, each year, many institutions do admit transfer students who have done well in college despite having less-than-stellar high school records or test scores. So it's certainly possible to "trade up" from a less selective school to one that is much more so. However, when it comes to the places you've named .... especially the Ivies ... transfer odds plummet significantly.
So my advice is to try to make the best of whatever college you plan to attend and do as well there as you can. Then, if you still want to move on, go ahead and apply as a transfer to your dream colleges but also find some other colleges that might be good fits for you but where the transfer odds are more realistic. Avoid going on "name brand" alone, and also keep in mind that, if you have a successful undergraduate career and stand out in the crowd, wherever you are, many doors will open up for you at grad-school time, perhaps including some that are closed to you right now.