I did martial arts for many years (among other sports). Then, I tried boxing and I was amazed at how fun it was and how much tougher it made me than anything I had done before.
Like you, I couldn't punch very hard when I started. But after learning the technique and practicing a lot, I became "heavy-handed" according to my instructor, which is about the biggest compliment I have ever received, athletically speaking. I can bench press maybe 135# on a good day…..but now I have a sledgehammer right cross.
Kickboxing is way overrated, IMHO. People do it because they either think they need to kick to get a good workout (women who want tight butt, etc.) or skinny guys like me who think they need to kick to have power. Most people don’t realize you use all muscles when boxing and you generate huge power from your legs and back. Best workout ever.
Don't be intimidated...just find a good instructor. Someone who will teach you and look out for you at the same time. Look for a gym that has other people with your same goals. You can work out with like-minded people and stay safe. I did run into a few punks who were dangerous to be in the ring with. But if your instructor is looking out for you, he will pair you up with safe sparring partners.
Edit: To answer your questions, I paid something like $50 per month for gym membership (unlimited gym time) and then $40 per hour session with the head instructor (worth every penny).