Driver ... then better wedges ... then a putter.
The extra yards off the tee leads to shorter approaches. Better wedges will let you stick more short approaches closer to the hole more consistently.
Then maybe think about a putter. That Starter putter served pretty damned well for a LOT of us for long time....is it a little inconsistent? Yes, but it will get the job done if you can learn to control the distance.
Starting with a better putter is fine.. a more precise putter will probably let you make more 25-footers, but it makes much, much more sense to try to avoid leaving them in the first place, no?