You don't necessarily need a smartphone. It is possible to tether to a dumbphone, if it has Bluetooth. Or possibly using a USB cable. Though would depend on whether the phone supports 3G/4G as to how fast it is.
If you want something separate, I'd get a 4G wifi hotspot. Then that should work with the Chromebook, without needing any special drivers etc.
For tariffs, maybe O2 PAYG? It is 1p per megabyte, not too expensive for light usage, no need to buy monthly bundles etc.