I have gone to the theatre, concerts, cinemas, and cafés on my own all my adult life, and never, even when I was very young, have I felt uncomfortable about it, or had anyone in any of those venues try to pick me up, when I was young and reasonably attractive.
As a young woman I would decidedly not have gone into a restaurant on my own, unless I could not find a cafeteria or a snack bar. Even now, at 70, I would not want to go into a pub on my own, but that is just me.
I can understand your hesitation if you have never gone to any of these places on your own, but I honestly cannot imagine that you need worry.