You can see pictures of Fermi surfaces online. Gold is a common one that is pretty simple. If you can find any 2-dimensional materials that would be cool since those are easier to visualize and might be simpler.
Reading about k-space and the 1st Brillouin zone would help you understand things better. If you encounter anything about "extended zone scheme", my opinion is that it is confusing, counter-intuitive and not a good idea. Ignore it. The way we do things, starting with atomic levels and letting each one broaden into a different band, it never comes up.
Graphene: I was thinking graphene might be a good example Fermi surface because it is 2D and should be easy to visualize. Then I realized that it is not really a metal, so it doesn't really have a normal k-space Fermi surface. In a way, the Fermi surface consists of 6 dots in k-space. See if you can find those online and understand them a bit. Then, if you were to dope it with extra electrons, the Fermi energy would move upward and each dot would evolve into a circle. So then the Fermi surface would be 6 little circles each one centered at a "K" point. If you can understand what a K point is for graphene in 2D, then I think you understand k-space and the Brilloiun zone concept.
I am thinking it would be interesting to cover graphene for several reasons: Easy to visualize because it is 2D. interesting material with interesting properties. of current interest... Also, it is an example of sp2 bonding. However, note that the band of interest is not part of the sp2 bonding, but rather it comes from the pz state of the carbon atom (which is one of the 4 orthogonal 1st-excited states of the attractive -1/r potential. pz wave-functions overlapping with pz wave-functions from neighboring atoms is what leads to the valence and conduction bands of graphene.
http://demonstrations.wolfram.com/GrapheneBrillouinZoneAndElectronicEnergyDispersion/
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