Yes, you can certainly get into a good grad school. I started a PhD in psychology in UCLA, but later moved to MIT to focus on mental health technologies. At both programs, I met exceptionally talented students who came from colleges and universities that weren't know for their psychology departments. Some of the grad students hadn't even majored in psychology!
The good news if you can major in almost anything and then pivot to a grad program in psychology later. The less good news - depending on your perspective - is that you'll like have to do some research internships before you start your grad program (especially if you want to go to one of the top graduate programs). Many people do these internships for a year or two after they finish college. I adored my research internships and I spent two years doing research assistantships after college before I even applied to graduate school.
Preparing for the psychology GRE will also be a great way to get caught up on the general body of knowledge you'll need to build upon as a grad school.
You can also pick up a lot of great books that provide sweeping overviews of different areas of psychology. I recommend:
How the Mind Works - Pinker
Thinking Fast and Slow - Kahneman
Influence - Cialdini
Unquiet Mind - Jamison
Learned Optimism - Seligman
Feeling Good - Burns
Full Catastrophe Living - Kabat Zinn
Rob recommends the following next steps:
- Find an internship or research assistant position. This will show the admission officers you are passionate about the field and you'll learn valuable skills you'll put to use as a researcher and grad student.
- If there's nothing that exciting at your own university, consider other locations. A research assistantship will pay a modest salary, but it should be enough to move somewhere else, have a small apt, and still have a fun.
- Contact the professors or lab managers at any labs you find interesting. Just tell them you're interested in getting more experience. Explain why you find their lab interesting and that you hope to get a research assistantship.
- Get books to help you prepare for the psychology GRE