Go to a good reputable local bike shop and talk to them.If you're really sure that a new bike will help then they can point you out to the right kind of bike and can fit you to something that will suit your budget. But take your bike and show what you're riding and what your problems are. Sometimes it's also just a matter of having them make some adjustments to your existing bike for a better fitting than you have now. I can't tell you how often I see people riding bikes that have the seat too low, the tires under inflated, the handbars at the wrong height and in the wrong gear, sweating and groaning and eventually giving up.

I don't know what model of bike you're riding and I know that riding a hybrid where it's hilly can be a lot of work and getting the mileage up can take a lot out of you. Just take it one step at a time. If you haven't done a lot of riding before, then it can be a lot harder. There are lots of beginner training programs out there that are quite modest in expectations that can be a good start for getting you bootstrapped with your existing bike.

Keep a log of your rides. Get a cheap bike computer to record your mileage. Once you gain more confidence you'll probably be buying more gizmos to track your performance.

I had an advantage over a lot of people because I was an ex roadie who had been off the bike for 35 years and I knew what to expect when I got back into it.
I worked with what I had for a year or so because I needed to figure out exactly what type of bike I really wanted and I needed to figure out how much money I should spend for what I wanted to do. I wouldn't recommend going out and buying a $3000 bike like I did unless they really knew what they were doing.

Step 1) Work with what you have. Get out and ride. Try to increase your mileage each time, and don't be hill shy. 90% of people I see out there are climbing hills in the wrong gear. There are plenty of websites that can tell you about proper technique. www.bikeradar.com in the UK is good. Go out every other day and try to stay out there a longer on the weekend. Make sure that you take a rest day in between.
Step 2 optional ) If you don't have bike shoes, get them, they don't have to be expensive (for now)/ Make sure that they'll take cleats for clipless pedals (that will come later).
Step 3) Get yourself a bike computer. Like I said, a cheap one will be fine for now.
Step 4) Keep putting on the miles. Nobody should be riding 7 days a week so make sure that you take a rest day.

Riding is something you should enjoy. Right now it seems like a chore but as you get into better shape you'll get a better sense of what sort of bike you'll really need.

For all dirt and lots of hills with loose gravel, you're looking at a mountain bike. If it's only going to be used for part of the year, then I'd look at a cheap hardtail because it sounds like a lot of your riding is going to be on the road.

One more thing, if you get yourself into shape on a heavier hybrid first, you'll be in excellent shape on a roadbike. I know a lot of people who are accomplished athletes who start their training regime in the spring with a heavy clunker for the first month and them move over to their high performance bikes after that.