I've rooted lots of begonia cuttings like that, starting in water until there are lots of roots, then into potting soil. So get yourself about a 6in.(or a deep gallon size would be fine) pot and a bag of potting soil. So you don't break off any roots when you pot it into soil, moisten the soil first, then put handfuls in around the cutting, and water it down so that it gently flows around the roots. Don't pack the soil down. Just keep filling the pot with soil until all the roots are covered, and maybe a little way up the stem. I do this in the sink, as some soil will flow through the drain holes. You can put a paper towel over the drain to keep the soil from clogging your drain.
This will probably get to be a big plant, so be prepared for it to need a bigger pot before next summer. You can grow it outside, somewhere shady once the nights are staying over 50 deg. or so. They do like to stay pretty moist, especially in hot summer weather.
