I've been on holiday twice on my own, both times to Tenerife (as I'd been there before).
During the day I walked, explored, had a few pool days, people watched.
In the evening I went for dinner on my own and would go to the waterside afterwards and have a wine while watching the sea (and people) before going back to my balcony.
I went self catering, as ironically I find eating alone in hotels more awkward than cafes, bistros and restaurants outwith.
I enjoy an early start / early night on holiday and I found going solo worked for me as I could just get up and go walking as soon as I woke without waiting for anyone, or having to go back to hotels etc when others woke. My day (and in turn holiday) was my own.
I enjoyed it and never at any point felt unsafe. Even with it being the tourist area, I found being alone avoided the "oh you wanting a drink, a good time, food" kinda rep people which I appreciated.
I recommend going alone greatly and now I know I can do it myself I'll definitely be working my way through my bucket list of places to go.
I'm very similar, in that I've been to the Canaries three times, on my own.
They were my first solo holidays and felt very safe. I would recommend a package holiday as a first solo trip because having a transfer to and from the airport is comforting, it was for me anyway. I'd say any resort in Spain, Portugal or Italy, would feel safe.
I think a week is long enough, when you're on your own, I think I would have felt a bit bored if I'd stayed longer.
I was in self catering apartments because I knew wouldn't enjoy sitting in a hotel dining room on my own.
I did get chatting to people when sitting around the pool, and was even invited out for the evening with them but I'm a bit of a loner, and if that hadn't happened, it wouldn't have bothered me. I was very happy relaxing with a glass of wine on my balcony, in the evenings
You'd also get chatting to people on the day tours sold by the resort, of course.
So, I think a package holiday is a great way to try solo holidays.
I used to work with a lady who loved going away on coach tours of Australia, NZ, and South Africa. They were really expensive but she always had a great time.
I've also had some city breaks to Glasgow and London. I love museums and also went to a theatre matinee. I felt that I had plenty to do and see, and three or four days felt long enough.
One place that I stayed on my own, for a couple of days, that didn't feel safe, was Los Angeles.
The public transport just didn't feel safe, and there were beggars who were quite aggressive etc.
Having been to New York and San Francisco, with friends, I would say you would be ok there on your own, they felt safer than LA anyway.