Welcome!
You're right. Core work is very important. I learned almost all of what I thought were saddle issues or numb hands issues had to do with a weak core early on. I now religiously do core work two to three times a week. It's made a huge difference in comfort as well as I look better, too. 
Team Haley, I was 40 pounds overweight when I started cycling, and almost all of it in my rear and thighs. Yes, they got a bit bigger at first because I was building muscle underneath the fattage, but I've really been focusing on diet, and my legs are starting to look damn good, if I do say so myself.
For me, cycling is about fun and health. I could only do three miles at first without thinking I was going to faint at first, and now I can easily ride 80+....well....not this week as I'm just coming back from an injury unrelated to cycling, but all of it was fun. I've met new friends, developed a kinder relationship with my middle-aged body, and just enjoy getting out there, like I'm a kid again.
Have fun, take things slow, see if you can find a couple friends to ride with or a club you can join. I've learned everything from riding with others, though I often ride by myself because I find it meditative and relaxing to go at my own pace, whatever that might be on any given day.
2013 Kirk Frameworks JK Special/Selle Anatomica
2012 Gunnar Sport/Brooks B17
2001 Calfee Tetra Pro/Selle Anatomica
1984 Raleigh Sport/Brooks B66