First I should say, that not only I went through the interview process when I was hired, but later, I when to multiple interviews as internal candidate to various GEICO teams, and I also participated in several interviews when we were hiring people for my current team. I can say that the process is not the same all the time, it depends on a position and on a hiring team. So I will describe you not my particular experience, but what you can expect in general. It will be relevant for almost any analytical position in the headquarters office.
First you submit an application on the website. Advice: if you are not a citizen, but you have a work permit, even if it is not permanent, check that it is permanent. GEICO will never help you with H1 visa, but if you have your green card, or J2 visa, or whatever which allows you to work in the U.S. then you can be employed and it will never be a problem. But if you don't check that box, then your application will not be considered. Another advice, if you see several interesting positions apply to all of them. It's often hard to figure out what will fit you most. If you have analytical mindset and some math and technical skills, you can be considered for any analytical position.
Then HRs screen your application, if they are interested they will send you a letter. Usually they will ask you to send them your resume, transcripts, cover letter, and fill out an analytical questionnaire. The latter usually asks about projects you did in the past, your goals for the future and so on, the questions which are often asked by phone in other companies. It's very convenient, as you can sit and think and write it carefully, much easier than a phone interview. And it prepares you for the next step.
The order of the next step will depend on a team, some teams do a phone interview first, and some teams do an online assessment first. Let's start with the online assessment. It is again depends on a team but usually it as logical test and a math test. Math test is very simple, logical is more challenging because of the time restriction. They will send you a link to the test. Notice the name of the company on the link, do your research and find their website, there you can find those tests for practice. It's a time worth investment, because with bad results you have almost no chance to be hired and I so some candidates about whom we were very positive, who failed the test.
You should expect to talk to 1-2 people on the phone interview. Very likely you will be given some analytical question to answer. Advice: There is no right or wrong answer, they want to see your thinking process. Don't stuck on one though, try to come up with different ideas and thoughts about the question. Don’t stress too much, people at GEICO are very laid back and friendly. From time to time they would ask for a second interview, it usually means that they are not sure if you're a right person. Make your notes during the first phone interview, think about it deeply later, and be prepare to sell yourself better if there is a second interview.
If the previous steps went well, then you'll be invited for a personal interview. (In some rear cases people were hired without ever meeting the team or even the manager personally, but it unusual). Usually you would have one or two interviews with several people (managers, team members, directors), but it depends. During the interviews you will likely be asked multiple analytical questions, they may show you charts and ask for your ideas about them. Again, they want to see your thinking process first of all, so don't be shy. Also on some teams you'll be given an analytical assignment, which you complete using Excel and Word. Those are not complicated usually, but there is time limit, and it may be hard to concentrate, so just don't stress too much and read everything carefully to not make stupid mistakes in calculations for instance. Then, usually you will go for a lunch with 1 or 2 team members but not managers. That's a good time to ask more about the work schedule, the office, life-work balance and so on. Also they will probably give you their business cards so you'll be able to write a thank you letter at this point.
Then if they decide that you are good fit for them, the HR will contact you and give further instructions. If you have any concerns, if you need to postpone you begin date and so on, my advice - write directly to the hiring manager. The communication between HR and managers is not good at GEICO, so talking to you future supervisor instead of HR is a good idea.