Plane tickets – use your favourite flight search engine. If you don't have a favourite, try kiwi.com.
Car rental – use the local CICAR car rental company. Don't rent a big car; you'll find it easier to park a small one and you'll be glad you had a small car on the way to Masca, for example.
Entrances and hikes – BEWARE some paths and places require prior permission for entry. Very often well in advance! This applies to entry to the Pico del Teide and subsequently to cable car tickets. Then the Masca trail and the Barranco del Infierno trail. They won't let you go anywhere without permits.
Prices at the destination – as can be expected, nice accommodation isn't exactly cheap. Otherwise, prices are similar to mainland Spain.
Calls and data – for EU citizens, everything is the same as at home. WiFi is common in hotels and restaurants and is free.
When to come – you can travel to the Canary Islands at almost any time of year. Just don't count on swimming in the winter months, not even in the heated pools.
Specifics of the weather – the Canary Islands are characterized by very frequent and strong winds.