Need help arranging travel from Edinburgh to Exeter to NY & Rome

Let's starts with the easy bits.

flybe.com operate between Edinburgh (or Glasgow) and Exeter

for Rome to Edinburgh ryanair.com and vueling.com

for Bristol to Rome easyjet.com

trains from Exeter to Bristol http://www.nationalrail.co.uk/

There may be easier options than Heathrow to NYC as Exeter to LHR is a bit of a faff.

You could easily fly from Exeter to Dublin, Manchester, Amsterdam or Paris to get back to NYC. So you may consider looking for multi city flights - into Edinburgh and out of one of those airports as alternatives to Heathrow.

Use Kayak.com and Skyscanner to help with your research

To find out who flies where nonstop from an airport, go to that airport's Wikipedia page. Here's the one for Exeter airport: https://en.wikipedia.org/wiki/Exeter_Airport_(UK)#Airlines_and_destinations

It shows that there are no flights on carriers that would continue to the US all on one ticket (for instance, British Air from Exeter to LHR to JFK does not exist, nor does KLM from Exeter to AMS to JFK). So, you'll have to take a train or flight the night before to a city that does have service to New York. Note: the night before, not that morning, just to make sure you don't miss the plane, since you'll be on separate tickets, and except for a few from London, all fights back to NYC will leave in the morning.

Of destinations served by flights from Exeter, there are nonstops back to New York from Amsterdam, Dublin, Edinburgh, Glasgow, London, Manchester, and Paris. If you want to sightsee for a day in any of these, this will work well. All of them have plenty to see to fill your one day; if you need tips for Glasgow or Manchester, may I humbly suggest my trip report from September 2016: https://community.ricksteves.com/travel-forum/trip-reports/uk-trip-report-glasgow-manchester-liverpool-in-september-2016

The trains from Exeter to almost anywhere with a flight to NYC, including to London, are long; in this circumstance, I'd fly. However, I see that trains from Exeter to Birmingham are less than three hours, and United flies from Birmingham to Newark, if that works for you. You'd still need to get to Birmingham the night before.

So, pick a city where you want to end up for a single day, then book open-jaw tickets, using the "multi city" or "multiple destinations" function of the airline websites. For instance, NYC to Edinburgh, then Paris to NYC, all on one ticket. If you want a roundtrip, you can use Edinburgh, since there's a flight there from Exeter. Then book your Exeter to departure city ticket separately; search on Skyscanner as said above, and book directly on the airline's website (most likely to be Flybe, since they have most of the service at that airport http://www.flybe.com/).

For your daughter, many of the flights from Rome to the UK will be on budget airlines. These have no discount for round trips, so she can book two one-way tickets for whatever configuration works best. Again, Skyscanner will show her all options, and she should book on the airline's own website.

Note that budget airlines make their money with extra fees for all sorts of things, and by having rules they enforce strictly. So, be sure to read all the fine print. If you're allowed one carry-on, that doesn't mean a suitcase plus a purse; they'll force you to check one, and pay extra since you didn't book a checked bag in advance. For Flybe, you can pay extra at the time of booking for a ticket that not only includes a checked bag but also ticket flexibility - very handy.

Great info. Thank you everyone!