In reply to Chris Sansum:
It's hard to know without knowing whether your speed is currently restricted by the phone or the network. Do you have WiFi at home? If so, how does the phone work on that? If it loads okay, your processor is okay. Some older low-end smartphones are really underspecced to cope with normal smartphone tasks, but most modern ones should be okay. I suspect the HTC One Mini would be fine. I'd also suggest you look at The Motorola Moto G 4G (low-end price, mid range performance) and the Google Nexus 5 (mid range price, high end performance)
Where do you live? What network coverage do you have (2G,3G, 4G)? If you have 4G coverage, or are somewhere likely to get it soon, then make sure you get a phone and tariff that allows 4G as it is massively faster and has not yet developed the congestion that besets 3G in some areas/networks.
3/EE and Voda/O2 share networks to some extent, but not completely. All are currently investing in 4G as well as other improvements so things should get better, whichever you choose, but try to get some first hand experience from locals.
Taking a contract isn't for everyone and with the rise of direct sales from the likes of Apple and Google it's not always cheaper to get the handset on contract either. There are some great deals around right now on Pay as You Go, so don't leap straight into a contract.
Virtually all the operators do inflationary price rises. It drives customers nuts, but with inflation as it is there's a strong financial incentive.