You've gotten some great advice from some who ride a lot more than I do, but I'll chip in because I ride a lot like you're planning to; average 45 minute rides that take me about 11 miles, with an occasional 30 or so mile ride.
I ride a Fuji street bike that I bought new in 1982 for $300. When it's in tune, it's still a thing of beauty. I do get passed by guys my age (late fifties) riding much more expensive bikes. They may be better riders, they may just have better bikes. I would love a new bike, because the easier it is to ride, the farther you can go, and the farther you can go (at any given fitness or comfort level) the more you will ride. And of course, the more you ride the more fit you'll get.
I have a slightly different view of two things that others touched on, that I think you should consider. I sit on a $200 seat attached to a $300 bike. It is not wide and soft, it's actually quite hard and thin, with a big hole down the middle, but I almost don't feel it at all. It touches my butt at just a few points and it barely touches me at all. I sit on an exercise bike at the gym and that big wide seat makes me stand and squirm and shortens my ride every time.
The second are pedals and shoes, which were mentioned briefly but I think being able to pull as well as push when you pedal also increase your comfort. You'll use a wider range of muscles and a smoother "spinning" style.