It must have been Napoleon who said, "time spent on reconnaissance is never wasted" it's the second rule of warfare. The first rule is, " don't invade Russia", that was General Montgomery in 1945, but I digress. I have spent a week doing some reconnaissance for you.
The best time to go is now, perhaps a little later if you want migratory birds or May for flowers, the weather is perfect, shirt sleeves all day, no heavy waterproofs needed, just a light windproof jacket at the bottom of your pack with factor 50 sunscreen to hand, it's the one your grandchildren use.
The roads are amazing, almost traffic free and no potholes so hire a small car. Even remote villages have excellent roads slightly wider than our single track roads in the highlands this is necessary as Cypriot drivers don't take prisoners when cornering. Drive on the left.
Troodos is probably the best base for a few days, two hours from Paphos airport and a hub for walks. When Cyprus was part of the British Empire the entire colonial administration moved up here in summer to avoid the heat down in Nicosia. The Jubilee Hotel is one of the last remnants of British rule and residence.
The trails are well engineered, they tend to contour around the mountains, are well surfaced so trainers will do and they are well way marked with a combination of sunshine and shade as they weave in and out of the pinewoods