If I go to LSU for a couple years, can I then transfer to Tulane or are the chances of that low?

Space considerations play a role as well as grades and the like. But if you do well in your classes, get involved in some activities, and have a good reason for wanting to transfer, you certainly have a chance. Probably what major you are choosing (since you will have a major by then) will also be a consideration. If it is one that is common at Tulane, that might work against you. If it is a more unusual or at least less popular major, that might be a plus. All you can do is try!