What is the process for setting up clinical trial visits? Who pays expenses?

Usually the trial. Most clinical trials are arranged by the sponsor of the trial (usually a pharma company or device maker) who wants to satisfy fda requirements for approval. In most cases, the fees are paid by the company. Some even pay you for participation. In some cases such as cancer therapy, insurance will add some payment. If you have some illness, you can google it and trials to see if you are eligible.