I do most of my travelling on my own (though granted it's not far, just Europe)! Really don't like the thought of having to coordinate everything with someone else. I did travel with an ex once and it was fine but only because we were so close & I felt totally comfortable saying ok, now I want time alone. With a friend I would feel like I was looking after them, sort of.
It's great, travelling alone...the spontaneity of it. You set your own timetable, eat when you want, see what you want, don't feel like you have to make sure someone else is having a good time, if something doesn't work out e.g. you miss your train, you just have to cope with your own reaction & not feel responsible for cheering the other person up. And a hotel room all to yourself
and the freedom, if you meet a handsome stranger 
I also think being put in tricky situations e.g. getting totally lost, feeling isolated, getting pickpocketed, missing flights, finding out your traveller's cheques aren't usable, losing luggage or dealing with complicated journeys etc has really given me a lot of confidence. I know that I am totally capable of problem solving because I've done it in the past, so when a problem comes up I don't dwell on feeling shocked but just try to find a solution. I've felt completely hopeless and cried in the street before and nevertheless it all worked out fine. One memorable occasion - having my purse stolen just before I was about to catch the Eurostar, had no cash on me, I called my mum up for commiserations but instead she roundly told me off for being so stupid as to have been pickpocketed
(it's an in joke now), then arrived at the station to find that the train had been cancelled & rescheduled for the next morning, still managed to get home to the UK the next day though.
I think my first time on my own I was going to Germany for a week to stay with people I'd never met & do work experience, my German was shit, and then via night train going to France for a month. It was a bit daunting - I was 14 or 15 - but yeah, the confidence that summer alone gave me was brilliant.
sorry about your breakup though, hope you're ok.