Snakebite punctures (so called because of the two adjacent wounds) are a common outcome from clumsy kerb hopping, and arise when the tyre hits the edge of the kerb so hard that the tyre deforms and the rim outers hammer the pinched tyre and inner tube against solid concrete.
The sure way to save your tyres when going up kerbs is to learn how to bunny hop, and the tyres won’t touch the kerb at all. As you approach the kerb, pull the front wheel up just far enough to clear it plus a little safety margin. That’s easy enough, but clearing the back wheel is trickier. The proper bunny hop technique is to stand up out of the saddle, throw your weight forward and twist up on the handlebars to bring the back end of the bike up. I’m not much good at that, so I tend to cheat and slip in a micro-endo, where I throw my weight forward and snatch at the front brake briefly to bring the back end of the bike up.
It’s important to approach the kerb at 90 degrees, as if you get it wrong and the rear tyre hits the kerb, then the bike will just kick up until the saddle hits your butt, which is easily recoverable. I found out the hard way what happens when you screw up an acute-angled kerb hop one wet winters day at the end of a muddy ride. I had SPD cleats clamping my feet to the pedals (they’re a bit like ski-boot bindings, and release when twisted) which makes bunny hopping easier as you can pull the back end up with the feet, but on this occasion I misjudged the weight of my bike - it had several extra pounds of mud hanging of it - and the back wheel didn’t quite clear the kerb. Instead, it slid along it sideways like a greased rail, dumping me hard on my side. Normally I’d get away with that, but on this occasion a pedal clamp didn’t release properly, and snapped my left foot off. It’s been bolted back on since, I should add.
The above techniques are more for the hardcore kerb hopper, there are safer ways to climb over them at lower speeds. Again, stand up from the saddle with knees slightly bent, pull up the front wheel to clear the kerb (as before), but then to get the back wheel over when going slow you can climb it. Lean forward to unload the back end of the bike and let it pivot about your knees and the wheel shouldn’t hit with enough force to do any damage. You could play it safe and bring the back wheel up very slowly, in which case you need to avoid stalling by throwing your weight forward against the handlebars with just enough grunt to get the back wheel over the lip. Sometimes it’s possible to pedal up a kerb lip from a stop, but often there’s not enough traction and the tyre spins.
Having full suspension does indeed help protect the tyres against rough terrain as the shocks absorb most of the impact from the big hits, but I wouldn’t recommend a full-sus mountain bike for city use, as they’re heavier, pricier, and much slower on tarmac than most other bikes. What they do best is go fast over rough terrain, much faster than a bike without suspension could manage, as the suspension eats up bumps that would otherwise throw off even the best rider. And the traction and grip is awesome. The price to pay for this is increased weight, a less stiff frame, slower speeds over flat terrain, and handling that feels washy. This is with a nice full-sus bike, the cheap ones are just so horrible to ride they’re not even worth their bargain price. Proper full-sus city bikes are available; they’re more of a road bike with lighter weight and limited travel suspension arrangements, and are a good compromise between speed and comfort.
Hope this helps. Alternatively, snakebite resistant tyres are available, but they’re a bit weighty.