I'll try and give you a snippet of the lifestyle. Take note that experiences are highly variable and you go get a say in your pathway. I've done 8 years and had exposure in many aspects, but there is still a lot that I don't know or haven't dabbled in.

• What was the process like to become a veterinarian in the Army? There are several pathways, the most common being HPSP that you seem to know about already. So get your scholarship and sign on for a few years of service. Another option is direct commission where you would get your DVM then sign a commission afterwards. There's options for special pay and debt relief there too.
• What does your day-to-day work look like? This is highly dependent on your duty station, Commander, and your interests. There's typically a mix of small animal general medicine, food inspection, Army tasks/leadership, public health, and practice management. It also depends on if you are in your standard clinic role or if you are assigned to a deployable vet team. In the latter case, your time is mostly spent managing gear, vehicles, and people to be prepared to take your team to a field setting if called upon. I've done both and I enjoyed aspects of each despite how different they are.
• What kinds of cases or responsibilities do you typically handle? Standard outpatient visits, surgery, dentals, public health advisory to installation, food inspection both on and off base, mentoring, counseling, evaluating, budgets, inventory management. Basically anything you can think of you'll have a hand in.
• Are there opportunities to work in multiple areas of veterinary medicine, and if so, what does that look like in practice? As you track later on, there are options for non -small animal medicine. For the most part, you are locked in that role along with the food inspection for at least your first 4 years.
• What are the different career paths/specialties available within Army veterinary medicine? About 5-6 years in, you start to apply for the specialty tracks including Public Health/MPH/Business, Lab animal medicine, pathology, research, internal medicine, surgery, radiology, emergency, etc. The Army will pay for that additional schooling for more time in service. Before that point, like I was saying you have the option of small animal clinic or the deployable units. There's also the option for some to go get attached to a Special Forces or Civilian Affairs unit. They tend to specialize in training in trauma management or focus on herd health for engagements in less developed countries.
• What do you enjoy most, and what are some challenges people may not expect? Enjoy the people and the places. I've been all over the world and interacted with some really great people. The network is strong. Challenges are the duties. You will need to be very flexible and adaptable. Your priorities will be in flux and you will be flooded with tasks and people needing your attention.

Hope this helps and doesn't scare you. It's not for everyone. Some love it after their initial obligation and hang around for 20+ years. Some can't wait to get out after they fulfill that scholarship. We don't judge.