I am attempting to extrude a cookie cutter's vertices that I made, along the z-axis, like so (note: this is the tutorial example):
But my one only comes up as this:
I am obviously still learning, but I believe I am attempting to have faces/sides in mine, much like the Mickey Mouse cookie cutter?
It looks like your end result has filled faces. Can you perhaps describe what your steps were to get you here?
Now to my approach. There are many ways it can be done.
I am a fan of geometry nodes and I am not concerned about vert count so I would use a Geo Nodes technique to solve this. First I would find an open-source, royalty-free version of whatever image I need to create (a Mickey mouse in your case) because we respect other creators.
Use ⇧ Shift + A and import images as planes and import the image.
Next create a geometry nodes tree and call it whatever you like.
We'll need to subdivide the geometry so we have enough verts to work with in the next steps. Typically I would hook up 2 subdivide mesh nodes as shown in my screenshot.
Drop in a Delete Geometry node. Connect a Color Ramp node to the Selection socket so we can fine tune our selection.
Drop in an Image Texture node and select the image we imported as a plane.
Now drag a connection from the Vector socket on the Image Texture node and connect it to the Group Input node as shown.
In the modifiers tab you will need to click on the button show below. It will change the display and allow you to select the UVmap so you have the correct vector coordinates.
Now you can drag the handles on the color ramp and begin fine tuning the selection.
Drop in a Realize Instances node at the end of the node tree. You need this node so that the geometry is realised if you need to apply the geo nodes modifier. I have a powerful system so I don't need to, but somebody else might. (Applying the geo nodes modifier will reduce the number of verts which Blender has to hold in memory. All of the subdivided verts which were created before the Delete Geometry node, still exist in memory even if you can't see them).
In my case, the image had some ghosting artifacts around its edges so it gave me extra edges which I had to manually remove (I'll explain that a bit further on).
Your image might be better and so you might be able to skip that step and instead just drop a solidify modifier in after your geo nodes modifier and you're done.
If like me, you're not done yet, then you'll need to apply the geo nodes modifier and then go into edit mode and delete all your loose geometry. When that is done, add a solidify modifier and you're now done.
You can remove the faces only (X) and extrude up:
But your walls won't have thickness. Unless I miss an easy way to do it (Offset Edges doesn't seem to work due to the topology) it will be much easier if you had a version of your object as svg with contour, then you could right click to convert to mesh:
And extrude up:
