There are very few nationalities that don't require a visa to enter Russia, mainly ex Soviet states such as Armenia, Moldova etc., diplomatic passports, some flight crews perhaps or during events such as FIFA. Good for you that you don't need a visa.

Moscow traffic is horrendous especially in the rush hours that you are arriving and departing and you have precious little time to see anything of Moscow. Take the Aeroexpress train from SVO to Belorusskaya station and the metro from there to Teatralnaya station and walk around the area which includes Red Square, Bolshoi Theatre, GUM department store then head back to SVO.

April might be a bit cold to sit outside, drink coffee and people watch. I've experienced snow storms in April. I used to go to the Cafe Kranzler at the Baltschug Kempinski, cakes are awesome and great views of St Basils cathedral.

good suggestions from pp.
The Orthodox churches inside the Kremlin wall are majestic. ( if you have the time.....

Thank you! This sounds like the basic plan! Hoping for no snow storms!

We are going to be there on a Thursday - Kremlin's day off, I think. Guess the visit to the wonders inside the walls will have to wait for another visit.

you can purchase and print Aeroexpres tickets on line...
saves time

Will do! Thanks! (and saves money - just checked!)

One more specific question: My husband has a thing for outdoor fruit and vegetable markets. Not really flea markets, but more food stalls, fruits/vegetables, spices, flowers. Does anyone have a recommendation for an outdoor market in walking distance to Red Square, on a Wednesday (not a Thursday as I previously wrote). Thanks!

there is an upscale food market inside the fantastic GUM
...you might find some fruit there .

There's a farmers market at Dorogomilovsky but it is not walking distance from Red Square, you need to get the metro to Kievskaya and walk from there, you don't really have time, if your flight arrives at 6am at SVO (is this domestic arrival?), by the time you clear immigration (you seem to think you don't need a visa, if that's correct, then you might go thru different line to everyone who has one so will be quicker.)
Dorogomilovo Market

There's another one at Tulskaya but again not walking distance. Sometimes you can find fruit stalls in Bolshaya Dimitrovka Ulitsa, at least there used to be some. It's common to see stalls here and there, in the metro underground passes or pedestrian underground passes and indoor markets.
A great food store is in Tverskaya Ulitsa and that is walking distance from Red Square, very famous, Eliseevskiy.

Eliseevskiy

Thank you, Odin and Danon! Will let you know how the trip goes! stay tuned end of April!

Hi! We hope that all members of this group are here now, in Russia, at the FIFA World Cup. If you watch the matches in Moscow, you probably use the metro. We would like to tell you more about Free Ride for football fans and other cool services available in metro