The S3 and S4 are both excellent phones. The price of the S3 should be coming down quite a bit now that it has been superceded by the S4 so you may be able to get a decent deal.
Since the S4 has struggled to meet sales expectations, it is getting a little cheaper (apparently sales were so poor, Sammy cut the price to networks 4 weeks after launch), but it is still Sammy's top-of-the-range phone, so it won't exactly be cheap.
If you are up for a top-of-the-line device, pretty much every review I've read rated the HTC One as the best phone of the moment. The build quality is amazing and the camera tech is better than anything other than the new Nokia.
If you're looking for a bit of bargain, then the run away favourite has to be the Nexus 4 - a premium spec phone for a very mid-range price. So cheap, you might want to get it contract-free. Awesome and
gets Android updates first.
Any of the phones above will be as fast as you need and a delight to use - I'm sure you'd be happy with any one of them. If money were no object, I'd go for the HTC One, however since I'm tight as a duck's derriere, I have a Nexus 4