Is it a tuberous type begonia, Shirley, or a fibrous-rooted type?

If it's either sort a frost will end your lovely display Shirley. When that happens, if a tuber, bring it in, dry it off, label it and store frost free. As you have asked about cuttings.....why not take some and try it out, if you have a green house? Generally speaking the fibrous rooted ones are not kept from year to year, as so cheap to buy as plugs, or in small pots, or as decorative plants, as you were given.

Many of the cane-stemmed, angelwing, or "dragonwing" fibrous-rooted Begonias make good indoor plants, as also do wax Begonias (B. semperflorens)--which are the more common bedding types. Tuberous Begonias need a heated greenhouse to overwinter well "in the green", so are usually stored as tubers.

Usually what happens to me , if I bring a plant that has been outdoors all summer into the house, is that the leaves turn yellow and fall off. I have no way of graduating this process. I can't even bring home a cyclamen, bring it indoors and keep it alive!!! I'm a whiz with orchids flowering indoors. If you have a conservatory you may succeed in following T's suggestion.

To move a plant indoors, do it when the indoor temperature matches the outdoor temperature, give the newly moved plant plenty of light--a coffee table 6 meters from the nearest window just won't do it, better right in a bright bay window--and keep it out of the draft from the central heating.