I started going places by myself when I was in my 30s with my dog. I went camping to nearby state parks and went hiking. Then, when I worked at a reservation center for a hotel chain, I started using the employee rate at the hotels to go to places I had never been to see plays and museums, etc. Had I not gone by myself or waited until someone went with me, I never would have gone or seen the things I have done and seen.
For me, I'm not one that has to travel with a lot of luggage or even one big bag especially if it's just a weekend getaway, so I can just use a backpack to carry what I want to take and don't have to deal with or worry about lost luggage or having to handle a lot of luggage or toiletries since most hotels nowadays have the little soaps and shampoos in their bathrooms.
It's okay to want to travel alone. It's an exciting time to go and do/see the things you've always wanted to do/see, especially if no one else wants to do it with you. It's a chance to explore the things that you find exciting for yourself and whether it's something you like or not.
For me, if your family and/or friends worry about you traveling by yourself, talk with them to let them know where you're going, how you're getting there and where you're staying and that it's only for emergencies. Any trip I've ever taken, I've let my parents know so they know where I'm at in case they need to get a hold of me in an emergency and I've let them know when I've arrived in town and at the hotel. Then again when I've left at the end of the trip. BUT I have never regretted any of the trips I've taken. I mean, how could I have regretted the trip to go see Carol Burnett in person or to see Cher or TSO in concert? Memories are better than regrets!