The work of a structural engineer can pose a major public safety hazard if completed by an engineer without the proper technical understanding. As a result, structural engineering is highly regulated industry through certification. With the proper preparation and experience, the process is very attainable, but be warned: it takes time and effort to complete.
Step 1: Attend a university with an ABET accredited civil or structural engineering program.
NCEES will only approve an ABET accredited degree if you want to proceed with the licensing exams on the quickest timeline. If you go to a school that was not ABET accredited, you can still become a structural engineer, but you may need to wait longer and have more engineering experience before you can take the P.E. or the S.E. I know many engineers who completed their studies abroad or studied (non-engineering) physics as an undergrad - they are successful professional engineers, but their path to the becoming a structural engineer took a little more time and effort!
Step 2: Pass the Fundamentals of Engineering Exam
The Fundamentals of Engineering Exam (F.E.) is a six-hour exam that you are eligible to take in the final semester of your (ABET accredited) undergraduate degree. If you are eligible and able, take it at this stage! It covers a lot of information from courses you took in undergrad, that you, frankly, won't use much of once you graduate and start working. I recommend getting this one out of the way ASAP.
Step 3: Get a Master's degree.
This is becoming less or a requirement for many firms, but when I graduated from undergrad, most firms were only hiring students with a master's degree. My advice is to only get a masters right away if you are sure you want to pursue structural engineering. These degrees are not cheap! But if you do want to be a structural engineer, I highly recommend it. I learned so much in my master's program that prepared me for my career. As a bonus, your civil/structural engineering master's degree can be counted towards your "engineering experience" for the P.E. and S.E. applications. (More on that shortly)
Step 4: Get Engineering Experience
When you apply to take the P.E., you will need to prove that you have been actively working on engineering work under the responsible charge of a licensed P.E. The length of time required before the P.E. application will vary in each state. When I took the exam in New York State, the requirement was 4 years of experience, but one year could be a master's degree.
Step 5: Pass the P.E. Exam
The national eight-hour exam that is given twice a year, typically in April and October, that constitutes the "P.E." in most states. In those states, if you pass the exam, you are officially considered a professional engineer. (Hooray!) In California, there are two additional state specific exams which require an additional application (and, of course, fees). These are the seismic and surveying exams. Each are 2.5 hours long and are offered on a rolling basis. You cannot be considered a professional engineer in California unless you have taken these exams.
Step 6: Pass the S.E.
To be a fully licensed Structural Engineer, you must pass the exam called the S.E. The timing for this exam will also vary from state to state, so be sure to check in the state in which you are looking to apply. This exam is sixteen hours long, given over the course of two days. Similarly, to the P.E., it is offered twice annually, in April and October. The test is split in to two sections: vertical and lateral. Each section can be passed independently, meaning that if you pass gravity but not lateral, you don't have to take gravity again.
Hopefully that gives you some insight in to how to become a structural engineer. It is certainly a lengthy and challenging process, but you only need to focus on one step at a time, and as I mentioned, with experience and preparation, I'm sure you'll find each step attainable. Best of luck!