Welcome to Mass.!! I don't live in Everett, I am from Bedford. Bedford is about 45 minutes from Everett in between Lexington and Concord. This is great area to go for a day trip if you want to get away from the city. Some attractions in my area are:
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http://www.bedfordrecreation.org/SBP2009.htm
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http://www.barefootbooks.com/story/about_us/our_stores/
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I have been to Everett but have not spent a lot of time there. It is small city but far as I know it is relatively safe. It is close to Boston (25 minutes) so that is great benefit. Some good Boston ideas for young children:
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http://www.bostonkids.org/exhibits.html
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http://www.mos.org/
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Hope this helps!! Good luck with the move!!