How can I learn to become an experimental physicist?
To start with, one has to become a physicist, and that is the goal of undergraduate physics majors, supply the basic understanding of physics up to the time of study, the experimental results and the theoretical models within which the results make sense. This is a serious job and it is what makes a difference between a tinkerer, i.e. somebody who plays with electronics or computer programs or ... and an experimental physicist.
Should I read some books on the subject (which books?)
As mentioned in the comments the experimental part needs apprenticeship . It is as if a doctor would be complete finishing the MD degree from books without internship to apply the knowledge. And worse, because most methods are written down in obscure papers and new methods appear at the rate of months.
Should I ask the head of the physics department or the professor in charge of the junior level if I can just hang out in the lab and work on my own experiments (what if they say no?)?
Labs have expensive equipment and amateurs will not be allowed to handle them without supervision, so I would expect the answer will be "no". You have to see if your university offers extra experimental classes as an elective . For example here in this university there exist such courses.
Otherwise you should decide what will be your graduate studies in physics, look at the research in you university on the subject and go and discuss with a professor that could be your advisor for a thesis there. It is possible that he/she could use help in the projects they have.
Is there some way of performing fairly interesting experiments at home for cheap (does anyone have a resource for this?)?
YouTube is full of videos with various home experiments, if all you want is some fun. If you seriously want to become an experimental physicist and there are no extra resources in the university you attend, spend your effort in establishing good grades in the courses available, in statistics computers and electronic courses and then apply for a graduate course in the field you want to study . You will learn by working on the project and the graduate courses necessary for it.
There are also summer student projects , as this one in CERN where students are set to work in experiments and machine development projects and computers. Good grades are ofcourse important.