The National Park on the Zim side costs £14/$20 as a non-resident (and trying to blag your way in is ill-advised -- I saw some tourists get caught short when they could not prove they were residents and the very unpleasant "Green Bombers" who hang out on border-posts and entry-points like this did not take kindly to it) -- the Zambian side £6.50/US$10. They each have their charms and it’s well-worth visiting both if you have time.
On the Zim side, enter either down the track from Vic Falls Hotel, over the railway track and straight on or down Livingstone Way from town. At the entrance, they’ll give you a map of the paths and the way is pretty well sign-posted. Turn left to head towards the small museum which won’t take too much of your time and head on for the larger-than-life Livingstone Statue at which point you have to tun round as you’ll hit the NP fence. Retracing your steps via Cataract view with a great (if somewhat dizzying) panorama of the gorge to get some feeling of the power of the falls as you see the water flowing over the edge at enormous speed, and make your decision whether to risk life and limb lurching down the slippery moss of "Danger point" (not that they’re trying to tell you something).
Heading on, you’ll come to perhaps the best view -- Zim Falls Bridge -- a highpoint of which may be to join forces with any locals who are hanging out to watch tourists paying good money to jump off the bridge and dislocate their spine.
At quiet spots, look out for wildlife (especially deer) among the profusion of leaves and ferns and bushes.
Crossing over (you have to come out, cross over the bridge -- take your passport and some cash for a £6.50/$10 Zambian day-pass visa) into the Zambian side, you’ll find the NP there gives quite a different perspective of the falls and rapids. You’re closer to the falls and get a better feel of their fury and power, as demonstrated by some of the great names for the individual rapids :(Truck Eater, Terminator, Washing Machine and, my personal favourite, Pearly Gates!) although to be honest they do rather blend into one after a while (particularly if, as when I went in April, you’re there at the end of the rainy season and there’s more mist and rainbows and thunder of rushing water than actual views of it).
Last major tip is to make quite sure you have a highly unflattering waterproof poncho with you and a couple of plastic bags for your passport, cash and particularly your camera. Might be worth splashing out (pardon the pun) for a disposal waterproof camera for the occasion.
Overall then, if time was tight, I'd spend my time and money on the Zambian side for variety and sheer natural power - however, Vic Falls town has more to offer (and is better placed) than Livingstone so you're likely to be able to justify a trip to both.