Think about that one thing that you're totally, head-over-heels obsessed with. That's a hobby-horse. Everyone has one (ahem, Perez Hilton) that's a little embarrassing to admit.
When Tristram introduces hobby-horses, he identifies them as a linked pair, almost like body and soul: "A man and his HOBBY-HORSE, tho' I cannot say that they act and re-act exactly after the same manner in which the soul and body do upon each other: Yet doubtless there is a communication between them of some kind … so that if you are able to give but a clear description of the nature of the one, you may form a pretty exact notion of the genius and character of the other" (1.24.2). Hobby-horses are not exactly a man's soul (women don't seem to have hobby horses in Tristram's world), but they're almost one step up from obsession.
If we're going to talk hobby-horses, Uncle Toby's obsession with building fortifications has to come up. This hobby-horse is so weird that Tristram freely wonders "whether he was really a HOBBY-HORSE or no" (1.24.3), our first clue that the toys aren't a simple symbol for an obsession or a hobby. Toby is so into fortifications that they begin to define his identity. And who wants to be friends with War Guy?
Then we've got Walter Shandy's penchant for arguing. The guy won't shut up, and it's not like he's arguing about anything in particular—anything will do. Minor characters have their hobby horses, too. Finally, we've got Dr. Slop and his ridiculous forceps. Keep those far away from us.
Surprisingly, Sterne is totally okay with hobby-horses. In fact, he flat out states that "so long as a man rides his HOBBY-HORSE peaceably and quietly along the king's highway, and neither compels you or me to get up behind him,--pray, Sir, what have either you or I to do with it?" (1.7.3) Tristram has nothing but good things to say about Uncle Toby, and, although he does seem to think his dad's a little cuckoo, he still respects Mr. Shandy's learning. Hobby-horses can actually bring people together, because one of the major tenets of Shandyism is that you have to accept people on their own terms. There's no sense in expecting Uncle Toby to be interested in abstruse theological arguments, and there's no sense in expecting Walter Shandy to care much about fortifications. But they can (usually) accept each other's idiosyncrasies—their hobby-horses, or, more powerfully, their very inner being.
So, what about Tristram? As he tells us near the end of the book, he's got one too: "my hobby-horse, if you recollect a little, is no way a vicious beast; he has scarce one hair or lineament of the ass about him—'Tis the sporting little filly-folly which carries you out for the present hour" (8.31.4). In other words—the very book you're reading. A hobby-horse about other hobby-horses? It's like a bunch of Russian nesting dolls.