How could I use remote events to give the player items?

I made you a script for that to work.

If you want the part with remote events (server-sided): First, you need to put an RemoteEvent into ReplicatedStorage.

Rename the RemoteEvent to BreakWood

You will also need some scripts. Put LocalScript into StarterPlayer > StarterPlayerScripts. Name it WoodBreakingClient

local ReplicatedStorage = game:GetService("ReplicatedStorage")
local UserInputService = game:GetService("UserInputService")
local player = game.Players.LocalPlayer
local cursor = player:GetMouse()

UserInputService.InputBegan:Connect(function(input)
    if input.UserInputType == Enum.UserInputType.MouseButton1 and cursor.Target.Name == "Wood" then -- Check if user left clicks on a wood piece
        player:FindFirstChild("Wood").Value += 1 -- Add 1 to the player's wood value
        ReplicatedStorage:FindFirstChild("BreakWood"):FireServer(cursor.Target) -- Fires server so the wood breaks for everyone
    end
end)

Now the server script (in ServerScriptService): We'll call it WoodBreakingServer

local ReplicatedStorage = game:GetService("ReplicatedStorage")

ReplicatedStorage:FindFirstChild("BreakWood").OnServerEvent:Connect(function(player, target) -- Runs when BreakWood is fired    
    target:Destroy() -- Destroys wood (it applies to everyone)
end)

Client-sided (it will apply only to you): Do the same as above. Again, put a LocalScript into StarterPlayer > StarterPlayerScripts. Name it same as above, WoodBreakingClient

But now, we're gonna use different code.

local UserInputService = game:GetService("UserInputService")
local player = game.Players.LocalPlayer
local cursor = player:GetMouse()

UserInputService.InputBegan:Connect(function(input)
    if input.UserInputType == Enum.UserInputType.MouseButton1 and cursor.Target.Name == "Wood" then
        player:FindFirstChild("Wood").Value += 1
        cursor.Target:Destroy()
    end
end)

This is all I could tell you. Hope this helps!