I've got a roadie, a full-suspension and a hybrid. Got back into bikes in my early 40s after years since my last 10-speed.
I'd been at http://www.floridastateparks.org/oletariver/
watching the mountain bikers and it just looked like too much fun. They rent them there (crappy bikes, but it got me on the trails for a first time) so one day with a cousin we rented two bikes. I had a smile on my face the entire time. I'd never been on dirt trails before with a bicycle. Very fun. I was hooked that day.
I bought the Giant NRS1 which had just received a bike of the year award. It's innovation was locking out the back suspension (when on flats, so as not to lose energy for foward motion by bobbing up and down when pedaling) thru frame geometry rather than a locking pin for the back shock. It's still a pretty damned good bike all these years later and I've only had to change out tires and shifters so far.
I got up on my skills to handle the technical trails but after a few serious falls--one that threw me about 15 ft--I started realizing my age and taking it easier on medium to easy trails. Also I enjoy double track (generally hardened dirt roads), especially the ones that head into the Everglades or other wilderness areas of Florida.
Getting back into biking, I started using my mountain bike on asphalt and though fun as an urban assault vehical, jumping curbs and such, I kept getting frustrated by the lack of speed so I wound up with my first roadie since I was in my 20s back in college.
Later I got a steel hybrid as a beach bike and also so guests could join me on some of the easier trails. Hybrids can do the double track and maybe medium single track but I wouldn't take one on anything near technical.
Here's my current stable. Just random web pics similar to mine but my Jamis is dark blue, not red.
Mtn Bike: Giant NRS1
full suspension, great specs, tubeless tires, hydraulic disc brakes (same brand as what Harley used at the time), auto rear shock lockout thru geometry. Very fun & capable bike.
Roadie: Jamis Comet
Jamis is generally thought of as a mountain bike company but I saw this and really liked the specs and they had a great buy at the time. Has good equipment and carbon forks front and back to reduce road vibration. Not high end by any means but a good middle of the road roadie.
Hybrid: Specialized Expedition
Steel bike so very sturdy but also sluggish. Not for serious biking. Great for school, beach, shopping (I've a rack on mine) & urban assaults. Not great for extended rides but good uprite riding position for days when I've a stiff back but want to ride anyway.
I'd suggest you rent a few different types of bikes, on & off road and see which you might enjoy before purchasing. Also know what types of trails are available to you in your area. You mentioned the asphalt but look also for off roading. What's especially nice about off roading is that it combines biking with nature hiking (though, you don't see near as much as hiking because you have to always be watching the trail). Also it's really nice to bike without worrying about motorized vehicles running you down.