Since you haven't had any guesses, I'm going to try to say something helpful. Sort of. There are actually thousands of begonias, and without a picture it's pretty hard to guess what you have. You could go to the website of the American Begonia Society and look at pictures and see if you can figure out what you have. Sounds like it might possibly be 'Little Miss Mummey', but I sure wouldn't take any bets. I'm not familiar with your climate, so hesitate to say too much about growing it outside, but if it isn't either excessively hot or cold, you could probably keep it very happy outside. Partial shade, some sun, probably, but start it in full shade and move it into more sun as it gets 'toughened up'. Sounds like it's doing quite well for you!

Sounds like you may have what is commonly called a cane or angel wing begonia. See the link attached. As the article notes, they've been around forever, it could be very difficult to identify the exact cultivar your friend gave you. But, that won't stop you from enjoying it (I love them). As for as summering outside, definitely. As Greenelbows1 says, start by putting it outside in the shade - after all danger of frost, as one frost will turn it to mush - probably Memorial Day for you. After a week, move to more sun, just make sure it ends up someplace where it doesn't get the hottest sun of the day from noon to 3 PM.

Now the bad news - if you summer it outside, it will most likely lose most of its leaves when you bring it back in for the winter. But, since they grow so fast and can quickly reach quite a size, I think of that as an opportunity for an overall haircut. Plus, you get lots of cuttings (usually leaves closest to the growing tips will not fall off) to give to your own friends.

- Steve

Here is a link that might be useful: Begonia Society - Cane Begonias

I hope you see this, this is a website I just discovered with scads of pictures of all kind of plants including quite a lot of begonias.

Here is a link that might be useful: Jearrard's Herbal

Thank you for the link to Jearrard's Herbal, Kelpie. Nice pictures and sometimes very amusing comments. I'll have to go back and look some more; only did the begonias and some aroids.