With such a comparatively young and genetically fluid breed, there's a lot of variance in appearance, degree of expression of breed-standard behavior, and even personality that makes this question somewhat difficult to answer.
Then again, change the wording of the question to "What is it like to have a cat as a pet", and the same would largely be true. Cats, dogs, parrots – each pet is an individual, and the best a breed specification can give you is a "ballpark of possibility", rather than a "spreadsheet of expectation".
Specifically, though, Savannah cats tend to behave like a "tomboy cat that grew up in a house of, say, Great Danes"… They're tough, they play and greet much rougher than most cat breeds, and most of them really do behave a bit like they have some dog in them somewhere – attaching to one or more specific family members, and following them/curling up with them everywhere. Even the Savannahs that aren't very "dog-like" are much more social, and less desirous of "alone time" compared to the average house cat. Like most dogs, they like to be at the party, rather than just observing it from a safe distance.
Also, they leap. Not jump, leap. It's really a sight to see… Basically, give up on hiding/securing anything "out of reach" of a Savannah, because there is no such thing. This is also an issue outdoors – in many counties of California, for example, the maximum allowed fence height (without additional permits/permission from neighbors) is 6 feet – which admittedly won't stop most cats, but won't even slow down a Savannah.
Speaking of laws… As a hybrid breed, Savannahs are often a little trickier or more of a hassle to house in certain cities, counties, and states. For a basic 'feel' for how animal control will feel should they ever pick up your crafty escape artist, check out http://www.hybridlaw.com/, but also be sure to double-check with the local authorities and/or vets – sometimes the law and practical understanding/execution of the law can get slightly out of whack, and while you can be totally in line with the rules – it's really hard to resurrect a cat that's already been mistakenly euthanized, or to successfully sue those responsible.
I've also never seen or heard of a Savannah that hated water to the extent most cats do, and I've seen firsthand one who would intentionally hurtle itself into a full bathtub, cannonball-style (cat-onball?).
Beyond that, from the Savannahs I've met, and what I've read online about the breed, there really isn't a lot of common ground that isn't common to all other cats. The main thing is, whichever idiosyncrasies your individual Savannah expresses, it will express them to the Nth degree, including less desirable things like aggression/hostility (very rare from what I've seen/heard[†]), marking/scratching/eating your stuff (somewhat, er, less rare, shall we say).
If you're a dog person who married a cat person, the Savannah might be the perfect compromise… And if you're a cat person, there's really nothing to hold you back (unless you're also a bird person). Oh, right, except, you know, cost[*].
[* Unfortunately, last I checked, Savannahs of verifiable breeding still run from $1000 for a 6th+ generation "SBT" cat, right on up to $20K and beyond for "F1" generation cats with 50+% Serval in them. Think about that a bit the next time you're scoffing at that otherwise-cute tabby at the local shelter.]
[† Combine the social factor with the leaping and "roughness", and then just try to leave that cat alone in an apartment all day – imagine a cat that is much more prone to abandonment issues, who needs a "strong leader" to stay in line… In terms of the worst case scenario, they can be like the cat version of a pit bull, just at a much much higher price tag. There are more and less naturally aggressive or needy Savannahs, but in any case let's just say it's probably not a great "first pet" for someone who isn't able to give it the attention and direction that it deserves. ]