Hi Cat Solo,
I found myself single some years ago and, like you, thought that it would be daunting to holiday alone. For a long time I thought that the travel industry just catered for couples ('price is based on 2 people sharing') and wasn't interested in single people unless they paid a huge single supplement for a dingy single room.
I subsequently came across JY and another singles company in 2010, and since then I've been on 7 holidays with 2 more booked this year. It has allowed me to see and do things that I never thought I would (see the sunrise over the Nile from a hot air balloon, visit Abu Simbel, see Table Mountain in Cape Town, visit the Cape of Good Hope, see a lioness with three cubs a few feet away in Kruger National Park, fly over the Grand Canyon in a helicopter....the list goes on).
As for where to start, I would agree with Jaya that you should pick somewhere you've always wanted to visit....for me that was always the Nile cruise. Perhaps you have an interest that could be catered for on a particular tour (eg natural world = safari, scenery/mountains = USA national parks...you get the idea I'm sure).
I can't advise you on the taster weekends because I didn't take that option, but I would say don't be too put off if your preferred destination doesn't depart from your local airport - JY may be able to offer you a connecting flight or you could always use National Express coaches to travel to the airport for a reasonable price (JY do a good deal with National Express which I've used on 2 occasions previously and will use again this year).
Booking that first holiday is always a little daunting, but it can be exciting too, so pick somewhere you've always wanted to visit, be that near or far, and go for it! You will never be alone unless you want to be and you will meet lots of friendly people. I still keep in touch with someone I met on that first Nile cruise holiday in 2010.
All the best
Mike