I think this is too broad to fully answer if asking about all saw types. Power saws I think can be grouped together for convenience and because the issues are fairly similar for all, but once you get to hand saws it's a whole different ballgame.
For power saws, I think the idea of 'proper maintenance' may vary somewhat depending on the person. For all power tools and the bulk of their users, this would mean only two things, cleaning and keeping rust at bay. Although it is possible few people have the inclination or skill to sharpen their own power-saw blades at home.
Assuming the blade doesn't have a rust-prevention coating applied in the factory keeping rust at bay is basic and simple. Oiling or waxing are the obvious routes, just as they are with most iron and steel in all other tools.
Cleaning is much more important with power-saw blades because the friction due to the high speed of the cutting in certain woods leads to gumming up in a way which is nearly absent when cutting by hand. Softwoods are the notorious culprit here, and a few resinous hardwoods are known for problems in this regard. Some sheet goods can be a problem too due to the type of adhesive used. The buildup can lead to the blade cutting poorly and appearing to have become blunt, while in fact it's just that the teeth need cleaning to free the edges and re-establish the proper clearance angle.
There are various products made for cleaning resinous buildup on saw blades. But the basic approach is either a solvent action (dissolves the resin) or a caustic action (breaks down the resin), which is why you can also do this effectively with an appropriate solvent such as kerosene, denatured alcohol, acetone or lacquer thinner* or with a suitable oven cleaner, one using caustic soda as the primary cleaning agent**. Some woodworkers report good results with other general cleaners including Simple Green, Goo Gone, Greased Lightning and even good old WD-40.
Not only is cleaning important for the blade to cut well in the first place, a clean blade has a much better quality of cut but more importantly keeping blades clean means they stay cooler during use, which gives a longer service life. As with all high-speed cutting edges, heat will begin to weaken them so anything that makes them run hot should be kept on top of.
* I would recommend you use the mildest solvent that will do the job rather than going straight to lacquer thinner, it most assuredly works but it's relatively toxic and very fast to evaporate. Plus it will dissolve a lot of paints so may remove important branding or other information printed on the blade. Try kerosene or denatured alcohol first.
** Caustic agents may attack some brazing so use with caution on carbide-tipped saw blades. Completely safe on any all-steel blade however. Do note that this will also very effectively remove most paints and coatings and of course there's a risk of chemical burns.
Obviously with almost all hand saws rust-prevention is important, cleaning less so since it's rare you'd need to do anything other than brush away a few bits of clinging sawdust.
But with hand saws the concept of maintenance will go beyond the above two things and must include sharpening, which for many dedicated users is a standard part of keeping up their saws. This subject is beyond the scope of an Answer to properly cover and even fairly lengthy guides elsewhere online should be considered just primers on the subject, given there are entire chapters in sharpening books devoted to the subject.
Instead of even attempting to describe it here briefly I'll just tack on a few links. First, a very basic overview to introduce the idea:
Sharpening and Setting Sawblades on Dieter Schmid Fine Tools
Sharpening Hand Saws on Workshop Companion
A Tool Kit for Sharpening Handsaws by Chris Gochnour on Fine Woodworking
Ron Herman: ‘Sharpen Your Handsaws’ from Chris Schwarz on Popular Woodworking
Last four on YouTube:
How to Sharpen a Woodworking Handsaw by Paul Sellers
Sharpening the Frame Saw by Paul Sellers
Saws Part 1: Saw Techniques & Sharpening a Rip Saw from Lie-Nielsen Toolworks
Saws Part 2: Sharpening a Cross Cut Saw & Setting the Teeth Lie-Nielsen Toolworks