What is the best way to become a horseback rider when you live in a city? The closest options are taking a 'lesson' for 2 hours and then pretty much you are left on your own. I think travelling across Mongolia on a horse would be cool!
I've just come from there. Something interesting to note - unless you go really touristy - Mongolians traditionally use wooden saddles, something I was NOT expecting to see :)
They and Kyrgyzstan share quite a lot of the same culture, including horse riding, and the nomadic / semi nomadic lifestyle. But at least the Kyrgs don't use wooden saddles ;)
In both Mongolia and Kyrgyzstan (and indeed in parts of Kazakhstan and Siberia) there were frequent horse riding opportunities - from a few hours to several days. I'd decide first what terrain you'd like to do it on (bearing in mind it can get quite painful in the saddle if it's new to you), and then find a country with that.
Both Mongolia and Kyrgyzstan are fairly cheap to live in, but the tourist activities in Mongolia struck me as being very overpriced. I preferred Kyrgyzstan for the CBT approach (Comunity Based Tourism). You can email the CBT and ask about tours or suggestions.
From what I saw, I'd suggest perhaps doing it from Cholpon-Ata just near Lake Izzyk-Kul. It's a beautiful region and has both options of steep rides or open plains. And the opportunities are certainly cheaper, and you can recover afterwards on the beaches by the lake :)
There's an Australian guy named Tim Cope who rode horseback solo from Mongolia to Budapest over three years around 2004.
He now does guided treks in the area that you could join for about $5000.
As one of the testimonials says: