That's pretty cool! It is a big model.
As far as paints, first, Model Master enamels have what you need.
For the colors, there's a number of options. The model claims to be based on the original ship of the class, DD-445 Fletcher. I've not read any reasons to question that. This suggests that you would paint her in her original colors, circa early 1942. Or her immediate sisters.
My favorite source for paint information is Snyder and Short, http://www.shipcamouflage.com/camouflage_database.htm
While I don't usually single source info, this database always seems to be correct, if sometimes it omits some of the more transitory measures.
Three basic choices.
Measure 12 Mod. This would be the most complicated. The verticals are three colors in a pattern, Navy Blue, Ocean Gray and Haze Gray. Colors 5-N, 5-O and 5-H. Model Master 2158, 2157 and 2156.
The decks are Weather Deck Blue, 20-B. Model Master 2159.
Gator makes masks for this ship, scale and Measure. I've used them for this Measure on a different ship and they work well.
http://www.gatorsmask.com/ddde.html
Measure 21. The simplest. Vertical surfaces 5-N Navy Blue. The decks 20-B Deck Blue.
Measure 22. Fletcher herself was not painted this Measure AFAIK, but some of her early sisters were. Vertical surfaces 5-N Navy Blue up to the lowest point of the main deck level (a horizontal line the length of the ship, and 5-H Haze Gray up above that.
The decks 20-N Deck Blue.
Whichever you choose, be sure to get at least four jars of each.
Black boot stripe.
For the hull I like Tamiya Dull Red in the spray can. It's easy to get that big area done without a lot of fuss.
I buy all of the Navy Colors from Scale Hobbyist. You won't find them at the LHS.
That's what I know, glad to help.
Bill