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Fairly new Blender user here. I'm using Blender Render for this next project of mine and I'm wondering if there is a way to make smoother shadows for these arches.

As you can see, within the arches, each block seems to get less dark, as opposed to a smooth shadow. It might have something to do with how it's built. It is three different pieces, the arches themselves, the block roof and some cubes that fill in the empty space.

Being new, I couldn't really figure out how else to do it, but any input you guys might have would be great! Thanks!

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A fast and easy way to get smooth shadows on different faces is to set your object to smooth shading, then enable auto smooth and set a proper value (I usually use 66, it works fine on a lot of cases) :

enter image description here

Smooth shading modifies the normals (properties indicating how light behaves on your mesh) in order to make your faces look kind of if they were a unique surface instead of sharply defined faces.
Auto Smooth separates the faces forming a certain angle (above 66° here), this way you can easily let your face be sharp when they need to be. But it's an automated tool, It might not be enough on more complex geometry.

If you need more precision, you can "force" the shading to be smooth or sharp. For that, with Auto Smooth enabled, just enter edit mode, select some edges, hit CtrlE and chose "Mark Smooth/Sharp" (note : in versions 2.79.x and 2.8x, these entries are in the "Edge Data" sub-menu).

Example here, I marked an edge as sharp on my cylinder:

enter image description here

And it's damn ugly. But it's for explanation sake!

If you're curious, I recommend you to check Blender's manual about normals.

About subdividing your mesh as mentioned by others, it completely depends on what you are doing, for what purpose, workflow, technical constraints, ... There are no absolute rules.
My advice is: If you don't need to add more mesh detail, don't add any.

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EDIT:

As L0Lock said, to smooth an object, you can simply click on Smooth Shading (in the Tools panel, on the right of your 3D View > Tools > Shading > Smooth) and enable the Auto Smooth option (Properties panel > Data > Normals > Auto Smooth and play with the Angle value).

My first answer was to use both Smooth Shading and Subdivision Surface. Why that? Because it won't only create a gradient shadow, it will also smooth your shape, which is generally what you want. In that case, no need for the Auto Smooth, but you will have to add some edge loops close to your angles to keep your angles sharp. That said you'll probably need to change your topology to make it work:

enter image description here

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    $\begingroup$ To get smooth shadows you only need to use shading tools like smooth shading. I would also use Auto smooth with a medium-low value (usually 66 works fine). i.imgur.com/Ruu8pzS.png Subdividing the mesh has nothing to do with that. $\endgroup$ – L0Lock Nov 17 '18 at 19:13
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    $\begingroup$ what you say is ok if you have a rather high-poly mesh, with the kind of topology he shows you will have a gradient but the silhouette will be angular, but you're right I'm going to precise this in my answer $\endgroup$ – moonboots Nov 17 '18 at 19:22
  • $\begingroup$ What I say is for no matter what you are doing. The question is "I'm wondering if there is a way to make smoother shadows for these arches". Mesh resolution is completely off topic. $\endgroup$ – L0Lock Nov 17 '18 at 19:25
  • $\begingroup$ ok you should add your answer then $\endgroup$ – moonboots Nov 17 '18 at 19:29
  • $\begingroup$ @L0Lock What you say is not true. You get shading artifacts on smoothed low poly meshes in Cycles. These are called terminator artifacts: developer.blender.org/T37814. So to get smooth shading you really need to crank up the subdivisions, moonboots approach is correct. $\endgroup$ – Jaroslav Jerryno Novotny Nov 17 '18 at 19:35

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