Red Square is awesome, rich, historical architecture and consuming.
If the truth be told the best time to see and appreciate the 'Beauty' of the Red Square is in early hours of the morning (in summer), between 0430/0600hrs (especially on weekends, not many people about).
Then you can take a stroll, take your time (no crowd), smell the morning fresh air (feel free), have a cup of coffee (the Mcdonalds) at the end of the square is upon 24/7. And Yes this part of Moscow is 'safe'.
Better still if you have the opportunity; you can drive around the 'Ring Road' and take photos too.
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