I won't repeat anything that Terry MacDonald mentioned about the Western Cape - he is correct in every detail!
Instead my question for you is this: what exactly are you after?
I understand that you said that you are after landscapes, and I think (correct me if I am wrong) you hail from the USA, so, really, my question is are you after the kinds of shots that might remind you of various parts of the USA, or are you after shots that truly place you in a unique location?
The truth is that really every province of the country (even including Gauteng which many would argue, and correctly too, that it is the least photogenic part of the country) could satisfy an avid landscape photographer.
However many parts of South Africa, it could also be argued, share a lot of similarities with parts of the USA. Yes, there are mountain ranges, rivers, waterfalls, ravines and canyons, immense plains, beautiful trees etc. But, many of those locations could also be mistaken for being somewhere in the USA.
My suggestion to you is to consider Namibia for pursuing landscapes - the place is utterly unique, and most of the images you would shoot just could not be readily mistaken for being anywhere else!
The whole country is really just an outdoor photographers wet dream - and that definitely includes landscape!
The Namib Desert, the mountain ranges, the mountain passes and canyons are all absolutely epic! The light just has to be experienced to be believed - understand that you will be accused of using a heavy hand adjusting colour in Lightroom or Photoshop! If you like nighttime photography and star trails etc Namibia is arguably of the very best locations on the planet due to the minimal light pollution!
Even the towns of Namibia are exceptionally photogenic in and of themselves! They are clean and tidy, generally very safe (especially when compared to South African urban centres), and exhibit very interesting German colonial architecture.
There is an abandoned diamond town called Kolmanskop not far from the port town of Luderitz in southern Namibia that is also any photographers wet dream. It is set up a bit like a museum, but the individual buildings can be entered and clambered about in in ways not possible almost anywhere else. The desert dunes have entered into many of buildings. All in all, in worldwide terms, Kolmanskop is a unique photographic experience all by itself!
Namibia is also one of the few locations in the world where significant wildlife exists out of National Parks and conservation areas. Wildlife is everywhere and you will encounter it while seeking out landscape shots and therefore the wildlife can often be incorporated into your landscapes if you are so inclined.
Namibia is pretty easy to access! For one landing in Johannesburg after an international flight the flying time to Windhoek (the capital of Namibia) is about the same as flying to Cape Town. Getting a hire car (I recommend a genuine 4WD vehicle that uses diesel) is simplicity itself and the photographic opportunities will be there just driving from the airport to the city! The international airport outside Windhoek is the only major airport I know where one will be dodging wildlife on the way to the city centre (about 15 miles)!
The reason for the 4WD is that apart from a very few major roads all the rest are unsealed and distances are long and so having a vehicle that is capable as well as having decent fuel tanks is important!
Accomodation options vary from sparse camp-sites to five-star accomodation. I would not suggest venturing into the very remote parts of Namibia by yourself with a hire vehicle but there are still so many truly spectacular locations that are well within your grasp!
So, yes, as someone born and bred in South Africa I can give you many options for South Africa, but I honestly believe you will be better served by visiting Namibia instead...
If you are interested feel free to email me through LuLa and I can help you plan a trip, hopefully in Namibia, but also in South Africa...
