Being alone, my specialist subject!
Definitely get out there and enjoy things and try things. You'll stop thinking "I'm alone and it's weird" very quickly.
Cinema, theatre and gigs. I've been going to these alone so long that other people get on my nerves a bit if they join me now!
I've also been to a few TV filming/radio recording things, if you live near a studio and this might interest you it's a fun and free way to spend a few hours.
I went to the races followed by a gig in the grounds a few years ago. The races bit felt a bit weird initially, mainly because I'd never been before and didn't have a clue what to do, but once I got talking to some of the betting guys and understood what was happening I had great fun.
If there's an exhibition I really want to see, then a gallery or theatre.
Classes, normally dance but occasionally a fitness session for something I've never tried, or a workshop.
I'd eat at, say, a burrito bar or a tea and cake place or something alone. But unlikely a restaurant with table service, not because I wouldn't enjoy it, but I'd feel bad about taking up a table unless it was deadly quiet.
I've also been on UK and overseas trips alone and it was mostly bliss (but can be expensive, of course),.
There are few things I find more relaxing tha wandering through a city on my own listening to some music, I love it.