Is there a way to smoothly CFrame a part from one spot to another?

You can use the lerp method of a CFrame to return a CFrame that is interpolated between the given CFrame and the "goal" CFrame. You can use it like so:

CFrame:lerp(goal, alpha) --CFrame is the start CFrame, goal is the end CFrame, alpha is a number between 0 and 1 which tells the script how far to interpolate. If alpha was 0, the returned CFrame would be the start CFrame and vice versa.

How to apply it:

Using that on it's own will only return a CFrame based on the alpha given at that time. You would have to use a loop to smoothly tween from one CFrame to the other.

I like to do it this way:

local Sine = function(x) --Sine will be the variable for the function. Another way of saying function Sine(x)
    return math.sin(math.rad(x)) --Returns the sine of the angle x. This number will always be between 0 and 1.
end

function TweenPartCFrame(part,goal,duration,alpha) --part is the part you want to tween, goal is the end CFrame, duration is how long it takes and alpha is the Sine function we made earlier.
    local X = 0
    if part:FindFirstChild("TweenIndicator") then return end --If there is a tween happening, stop the function.
    local tweenIndicator = Instance.new("BoolValue",part) --Add value to the part to show it's being tweened.
    tweenIndicator.Name = "TweenIndicator"
    local start = part.CFrame --Variable of the start CFrame so we can use it to tween the part's CFrame properly.
    while true do
        local newX = X + math.min(1.5 / math.max(duration,0), 90)
        X = (newX > 90 and 90 or newX) --Sets X to 90 if newX is over 90, otherwise it's set to newX.
        part.CFrame = start:lerp(goal, alpha(X)) --Use lerp from the original CFrame to the goal CFrame with the value returned from the sine function.
        if X == 90 then break end --Breaks the loop if X is 90. We stop at 90 because the sine of a right angle will be 1 so we don't need to tween the part anymore.
        wait()
    end
    tweenIndicator:Destroy() --Destroy the value in the part to show we're not tweening it anymore.
end

--Example Usage:
TweenPartCFrame(workspace.Part,CFrame.new(),2,Sine) --Would tween the Part in workspace to CFrame.new(0,0,0) in two seconds.

I hope my answer helped you. If it did, be sure to accept it.

Well, if you want it to particularly move somewhere and be able to detect being touched, then you can try adding a velocity to the part. All you need to do is to set the velocity with Part.Velocity as a Vector3, then multiplying the Vector3 by how fast you want it to go in terms of guns.

Part.Velocity = Part.CFrame.lookVector * 150 -- This makes the part head forward at a speed of 150.

Otherwise, you can try doing a loop.

for i = 1, 50 do
    wait()
    Part.Position = Part.Position + Vector3.new(0, 1, 0) -- Makes the Part go 50 studs up.
end