What should I expect to experience during post-transplant recovery?

Depends on organ. It all depends on the organ being transplanted. But all pos-transplant recipients will be on an anti rejection regimen which can cause a variety of side effects.
Many things. Initially you will have pain from surgery. You will also experiene side effects of the medication. Some could be mild like nausea & GI upset or more progressive like tremors & problems consentrating. You will be more likely to aquire infections and those could quickly become severe. You will still have a lot of blood tests. More specific recovery issues would depend on what type of transplant.