Best and efficient way to store rarity of an object?

You can do both at the same time. You can do this manually, but it is error prone; use functions to automate it instead (below).

Also note that whenever you are thinking in terms of searching through a list to match a specific string, use a dictionary.

Code:

local function RarityGroup(name, color)
	return {
		Name = assert(name, "Rarity groups must have a name"),
		Color = assert(color, "Rarity groups must have a color"),
	}
end
--Rarities
local common = RarityGroup("Common", Color3.new(1, 1, 1))
local legendary = RarityGroup("Legendary", Color3.new(1, 1, 0))

local items = {} -- List<item>
local itemsByRarity = {} -- Dictionary<rarity, List<item>>
local function Item(stats)
	assert(stats.Rarity, "All items must have a rarity")
	local item = stats
	items[#items + 1] = item
	-- Get (or create) the list of items for this rarity level
	local list = itemsByRarity[stats.Rarity]
	if not list then
		list = {}
		itemsByRarity[stats.Rarity] = list
	end
	list[#list + 1] = item
	return item
end
--Items
local dog = Item({Name = "Dog", Damage = 50, Rarity = common})
local dragon = Item({Name = "Dragon", Damage = 200, Rarity = legendary})
local cat = Item({Name = "Cat", Damage = 50, Rarity = common})

--Demo: items by rarity
local commonItems = itemsByRarity[common]
for i = 1, #commonItems do
	print(commonItems[i].Name, commonItems[i].Damage)
end
--Demo: can get rarity name and color from any item:
for i = 1, #items do
	print(items[i].Name, items[i].Rarity.Name, items[i].Rarity.Color)
end

-- Say you want to look items up by name (ex maybe you store the name of an object in the datastore):

--Manual version
local itemsByName = {}
itemsByName.Dog = dog -- equivalent to: itemsByName["Dog"] = dog

local name = "Dog"
local item = itemsByName[name]
print(item.Name, item.Damage) -- will print Dog 50

-- You could expand on the 'Item' function to add items to the 'itemsByName' dictionary automatically

(Note that the assert commands are just there to make sure you don't forget information when adding groups or items.)

Doing things this way is a bit more complicated to set up (as you can see) and will use slightly more memory (though the difference will be very very little - you won't notice), but lets you look up whatever list you need without going through all your items each time.