Driving from Lusaka to Livingstone is straightforward. Most of the road is in reasonable enough shape, though last time I did the journey (about 4 years ago), there were still a few bad, potholed filled, stretches. But, even a regular (non 4x4) car will be able to do the journey fairly easily.
There are several very reasonable hotels and guesthouses on the Zambian side, so for that you will not have to cross the border. And unless Victoria Falls (town) has gone through an economic upturn, the nicer restaurants (and South African chains) are on the Zambian side.
You do not need a visa for the other country (Zimbabwe if you're coming from Zambia) to get onto the bridge that connects the two. But, if you're continuing onwards, whether you can get a visa at the border depends on the passport you hold.
Crossing from one to the other, though, is straightforward. But in your case, you will need a day visa for Zimbabwe and a multiple entry visa for Zambia. For many nationalities, these can simple be obtained at both borders.
It's said that the Zimbabwean side of the falls is prettier. In my opinion, this might indeed be true, but only marginally so. There's no real need to check out the falls from the other side if you think it's a bit much of a hassle.