I am wondering if anyone has advice on how best to get from the airport to our hotel in Moscow and then from the hotel to the other airport (into Sheremetyevo and out of Domodedovo).

Which hotel? what time of the day?

Depending on your answer, you'd probably be choosing between Uber and public transportation (aeroexpress then metro).

I used aeroexpres train ( both tickets on line ) and metro from and to Sher.,.,
but my hotel was minutes from the metro stop.
From Dom .I took a taxi to my hotel but I negotiated the price ahead of time
( I speak some Russian) which is what is usualy done.
Never used Uber.
The traffic in Moscow can be huge so metro and train might be more efficient.
Metro connections are excellent ; station names and signs are in the Russian alphabet.
The passport control entering and exiting the country took a very long time..
Because of FIFA some adjustments might have been made.
I was in Moscow in the summer of 2015 and 2016.

Without knowing which hotel you have booked or at least where it is, (you mention on another post as it being out of town), it's hard to come up with a suggestion. Moscow traffic is a nightmare at the best of times.

The hotel is the Pogosti.ru na Alexeevskoy located at: Ulitsa Malomoskovskaya, 21,str.1, 127562, Moscow which means nothing to me, but it looks like it is out Prospekt Mira in the Alexeyevsky district.

I wouldn't call Prospekt Mira out of town; whilst not in the city centre, it's not that far.
You can take the Aeroexpress from SVO to Belorusskaya station and then the metro to Prospekt Mira and walk or go onto Alexeyevskaya station if its too far to walk. If you don't fancy doing that, you can prebook a taxi from SVO to the hotel or use one of the official ones on arrival.
http://www.svo.aero/en/directions/

There are months before you go....take some time to familiarize
yourself with the metro and city map...
With a unregulated taxi service and the size of the city, it will help to have some idea how to navigate the metro.
If you can , purchase tickets to the Kremlin ahead of time...the lines are
huge and will probably be longer with thousands of visitors.
I also purchased tickets for Bolshoi on line (directly from their webpage.)

There are three ways

1. Public transport - the cheapest.
From Sheremetievo - Aeroexpress train to Belorusskiy railway station. And than - metro.
https://aeroexpress.ru/en/aero.html
To Domodedovo - metro to Paveletskiy railway station and than Aeroexpress train.
2. Taxi.
Strongly not recommend use taxi driver, that you can meet in arrival hall in airport, or in city. "Official" taxi - very expencive. Use Uber, Gett!
3. Private transfer. For example - from Sheremetyevo to center €25 https://transferdesk.ru/en/sheremetyevo-airport-taxi, and from hotel to Domodedovo €28.