Which carrier are you on? Which are you going with? Which area are you in? What do you want to pay monthly?
SIII is a great choice, but the Verizon version has a locked bootloader, which means no chance at custom ROMs like CM9/10. Area is entirely important as well. If you live in an area with LTE coverage, it may be worthwhile to go with Verizon if they have the best LTE coverage in your area. I don't really know the budget carriers in the US, but they have WAY better monthly rates than the big carriers, and if you're going to be in the city most of the time, the coverage area is not an issue.*
The difference between budget phones and flagships is minimal, it all comes down to the 'purchase price', so as long as you're able to drop ~$200 initially, do it. Do not get stuck with a mid-range phone that will be completely outdated a year from now. Go big or go home. Best out currently: HTC One X, Samsung GSIII, Motorola Atrix HD/RAZR HD(soon), and iPhone 4S(5 soon). Do not buy an LG, they make good electronics, but their phones are awful at the moment, however they are coming out with a 5" phone(huge) with full 1080p(better pixel density than Retina). I'm not familiar with Sony phones, they look nice, but I haven't heard any rave reviews.
*For example, I pay ~72/month on Bell for 1000min, 10 unlimited numbers, early nights, 1GB data, VM/CD/etc., but if I switched to Wind, it would be ~$40/month for unlimited calls/data. I will be switching soon, hopefully someone will take over my promo Bell plan, if not, even the cancellation fee is worthwhile.