Those pebbly beaches are on the west of the island (Pafos)
Ayia Napa has Nissi beach & Protaras has Fig Tree bay & they are beautiful, the best beaches are in the east, oh & the sea is calmer & better for swimming.
Pernera is a small resort just outside Protaras & it could be just what you are looking for. Cyprus is a bit pricey agreed, but eating out is not too bad, lots of restaurants do set meals for dinner, 3 courses for £6-£9, some even include a glass or caraf of wine with that.
Drinks can be pricey, but i'm almost tee-total, so not a problem
Personally I don't like Pafos or the west coast nearly as much as the east side.
The bus service is also very cheap & regular, you can do cheap trips to Niccosia (the capital) and Northern Cyprus from Protaras/Pernera & the bus will take you from Nissi beach, to Ayia Napa, to Protaras, to Pernera & into Paralimini for 50p!!!!! Its great value.