You didn't say how many gears on the rear, I presume 7, and possible a freewheel as opposed to a cassette, so this may also entale a wheel upgrade 7sp shifters can be had on
ebay from about £30. This is probably the cheapest option
Depends what the budget is.
Claris (8sp) or Sora Shifters (9sp), you will need something to replace the levers you currently have (adjuster barrels)
You should just be able to replace the chain rings, though the front mech may need moving up the tube a little, and gear cable if you buy 2nd hand shifters.