Best way to hit Venice, Puglia/Basilicata, and Sardinia (San Teodoro area) - in 10 days?

That's not enough time for all 3 locations which are at the opposite sides of Italy. Assuming you have 9-10 nights on the ground, you could do:
3 nights in Venice
6-7 in either Puglia/Basilicata OR Sardinia (can't do both as each needs at least a week to see).
You can also do in the opposite order.
Given the distances flying between these locations is probably the best option.
Volotea flies between Venice and Bari, and seasonally Brindisi too (both in Puglia).
Easy jet flies (summer only, not sure when you go in Sept) between Venice and Olbia in Sardinia.
You will need a car in both Sardinia and Puglia as using public transport to visit those regions would be time consuming and inefficient. Obviously the car is useless while in Venice.
None of these airports have direct connection to IAD, so you will need to connect via some other hub (like Rome FCO). Venice has direct connections to JFK, ATL (Delta), EWR (UA), PHL (AA), and possibly others next year, all are seasonal flights except JFK (all year around). All other airports in both Sardinia and Puglia have European flights only (mostly seasonal, summer only, with low cost airlines).

Many folks have said they try to avoid flying back to the US from Venice because departures tend to be very early in the morning, and getting to the airport from Venice is time-consuming and potentially very costly. So check schedules back to DC very carefully before placing Venice at the end of your trip.

You can explore flight options from Venice to eastern European destinations on skyscanner.com, though I wouldn't want to add yet another flight into your itinerary. I took a quick look and found non-stop flights from Venice to Budapest, several cities in Romania (I wouldn't do Bucharest), Split and Dubrovnik in Croatia, Berlin, Prague, Sofia in Bulgaria (not my favorite in that country), and Krakow and Wroclaw in Poland. Be sure to check on flight connections back home from the eastern destinations you are considering.

I'd spend all the time in Italy, myself. If that just won't do, I'd take the land route to Slovenia (Ljubljana) or the Istrian Peninsula (to get a taste of Croatia).

Feel you can only do Venice and (my choice) Sardinia. You can fly from Venice to Alghero on the West coast for around $108 one way from Venice in September(see Skyscanner). Of all the cities we visited in Sardinia, this one was tops for us. We were there in June, and the city was hopping into the late hours. Plus, if you rent a car, there are world class wineries to visit on the West and North coast. I don't have any feel for your original area, I just know when we researched our trip, we stuck to the West coast, based on what we wanted to do and what my Italian friends' advised. Finally, you won't meet many Americans here, it's really way off most U.S. traveler's radar.

These are great responses - MUCH appreciated!! I do think we will end up doing Venice and Sardinia and saving Central Italy for another time so we can really explore it properly. What are thoughts about trying to do Alghero and then the San Teodoro area? San Teodoro was suggested by a friend as a more affordable laid back spot with good beaches, etc....but I want to make sure there is enough to see and do there? Also - where in these areas would be best to stay?

I generally go to the Gallura (northernmost area) and stay in or near Santa Teresa). Any place on the coast from Alghero to San Teodoro is good really (not Olbia, which is a port city). The beaches are wonderful everywhere. Just be aware that the Costa Smeralda (Porto Cervo, Porto Rotondo, etc) is where the rich and famous with their huge yachts go, so prices aren't budget in those towns. Make sure you take a boat excursion of the Maddalena archipelago.