A few weeks ago, in a local thrift store, I found a Finnish leather-bound bible from 1901 - with the illustrations by Gustave Doré - so was naturally curious to know the answer as well, and did some digging.
Especially with leather books, it's important to keep it away from moisture and (direct) sunlight (this turns out to be the mantra of conserving anything art-related :). An atmospherically stable environment is desirable to prevent the leather (and the rest of the materials of the book) from shrinking and expanding.
Dusting should be done carefully with a dry cloth, a feather duster, or "a hairbrush sold as baby’s first".²
You mention this in your post, but I'll repeat it for the sake of completeness: when handling the book, preferably wash your hands to get rid of the most aggressive contaminants. Nevertheless, it's desirable to use bare hands, as the natural oils of our skin will actually benefit the leather and improve proper handling.²,⁴
Covering leather in plastic to protect it is discouraged, as it will hinder the breathing of the leather.
"Always avoid solvents and abrasives when working [with] leather."²
Be extremely careful and skeptical about using "DIY" products that are recommended by people without any related professional knowledge. What may result in an appealing appearance now may end up ruining the leather (e.g. Vaseline or glass cleaning products, and even saddle soap might not be as benevolent as some make it out to be ¹,³).
The following general rules on book-handling are even more important for leather-bound tomes: don't stack them too loosely or too tightly, as the first will cause the books to sag which might fatigue or tear hinges, and the latter will put strain on the hinges.
When taking a book off a shelf, never pull it by the top of the spine, but find a way to grab it in the middle.⁴
If the leather of your book ever becomes brittle or cracked, or starts to suffer from red rot, you can use something called Cellugel, which will stabilize the leather without discolouring it. This product has been recommended to the author of the article on biblio.com by a conservationist,² and is mentioned by another author³ as well.
Another product with similar effect suggested by the latter author³ is 'Restoration Leather Conditioner', and according to the author is used by "such institutions as the British Museum and the Library of Congress". The use of 'Restoration Leather Conditioner' is recommended on a yearly basis by another author⁴ to keep the leather in good condition (see this article for a guide). Do note that at this point where the leather and binding are new it might not be necessary at all to treat the leather yearly. I suggest consulting a conservator.
I will not go much deeper into treating damaged leather or red rot here, because hopefully it will be very long until your book suffers that fate. Apart from the fact that those cases should be consulted and probably treated by professionals, it might be an interesting topic for another Q&A.
Written by a now retired restorer of (leather bound) books
Written by the authors of 'The Care and Feeding of Books Old and New: A Simple Repair Manual for Book Lovers', Bern Marcowitz and Margot Rosenberg
Written by the columnist on book repair for BookThink.com Gail Altman. I cannot find any professional information on this author, however.
By Joseph Adams, who "has over 19 years of book binding and restoration experience"