how do I check if a player said something but also included a players name?

you can use string.match(string,pattern,int) to find a specific character, word or letter in a string. Also its suggested to use the chatted event on the server, instead of the client. use string.sub(string,int,int) to sub a string.

--server

game.Players.PlayerAdded:Connect(function(player)
    player.Chatted:Connect(function(msg)
        local lower = string.lower(msg) -- turns the letters inside the string to lowercase

        if string.match(lower,":fly ") then 
            local sub = string.sub(msg,6,100)

            print(sub) -- prints of what you could say "after the command"
        end
    end)
end)

if I chat ":fly zValerian" then it'll print "zValerian" in the output.

string manipulation

When parsing a string, it's important to define some rules in how your text is structured. In your case, you have a command prefix (what comes before the command), a command operator (the type of command), and a command operand (the arguments of the command) -- in that order. The command prefix is static, so it's pretty easy to check for that case:

local command = ":fly person"

-- if input starts with the command prefix, then...
if (command:sub(1, 1) == ":") then
    ...
end

You know that the characters after the prefix will be the type of the command, separated by a space, then followed by the operand (the effected player's name). Let's look for the type of the command first.

local command = ":fly person"

-- if input starts with the command prefix, then...
if (command:sub(1, 1) == ":") then

    -- find the characters that come after the prefix, and before
    -- the separating space character.
    local commandType = command:sub(2, "%S+")

    -- if the command type is "fly", then...
    if (commandType == "fly") then
        ...
    end
end

In the code snippet above, I'm using the character class %S to represent anything that's NOT a space character. The + quantifier just tells the program that I want all characters that aren't a space character, up until the first space character it finds.

The next sequence is similar; I will use the same string pattern as I did before, but I will use it only on the end of the string. This can be achieved by using the $ character, also classified as a magic character, that will apply the string pattern at the end of the string.

local command = ":fly person"

-- if input starts with the command prefix
if (command:sub(1, 1) == ":") then

    -- find the characters that come after the prefix, and before
    -- the separating space character.
    local commandType = command:sub(2, "%S+")

    -- search for the command argument starting from the end of the string
    local commandArgument = command:match("%S+$") 

    -- if the command type is "fly", then...
    if (commandType == "fly") then
        -- do some method on the commandArgument (the player) that makes them fly
        doSomeMethod(commandArgument)
    end
end

I'll edit this answer a little bit to include some useful resources where you can learn more about string patterns and methods. Until then, I hope this answer was somewhat useful. Let me know if you have any questions.