The good news for travellers is that from April 2016 EU "Roaming" prices are going to drop considerably and as from June 2017 there will be no more roaming charges within the EU. Unfortunately this is not a world-wide decission so countries not in the European Union will continue to charge what they like. Currently phone companies are allowed to add 19 cents a minute to your outgoing calls, 5 cents aminute to incoming calls, 6 cents for a text message and 20 cents for ever MB of date used. In April 2016 those priceswill drop to 5 cent for calls, 2 cents for SMS and 5 cents per meg of data. Already a good deal. Then in June 2017 you pay the same in any EU country as you would back home.
The two main companies with their own system are Movistar and Vodofone/Ono. Both companies are installing their own fibre optic system to provide land-line speeds of up to 300MB. There's strict laws about setting up new telephone masts so currently almost all are owned by Movistar and the other companies are able to use them too. As far as landlines go, the other companies (other than Ono) all use Movistar.
When I first arrived here in 1998 there really was only Movistar but the competition has grown quite fierce over the last couple of years and every 6 months or so new offers come flying out.
So, if you're planning on using the Internet from your mobile device then the best option after wifi is to get one of the prepaid sim cards from one of the operators here that have a shop you can buy from. The main ones with their best offers as of summer 2013 are :-
Movistar - 3 euros for the card then 7 euros a month for 1GiG of internet plus free local calls
Ono - I can't actually find a pay as you go option!
Yoigo - 10 euros for 1GB
Vodaphone - 9 euros for 1GB and 20 minutes local calls
WiFi Canarias are offering for 30 euros 1GB internet but in the form of a mini, portable router that lets you connect up to 5 devices at the same time.
4G has arrived by the way which will see mobile speeds increase to 20MB and more per second.
Staying here longer than a holiday? As far as landline services go I would check with Movistar and Ono first to see if either offer fibre optic. Cable speeds currently reach a maximum 10MB but fibre optic are offering 30MB up to 300MB. Both Movistar & Ono are currently offering a package for 50 euros a month of Landline with free national calls plus 100 minutes to mobiles, a mobile phone line with 200MB calls and 1GB fast Internet useage a month and 30MB per second Internet at home. For an extra 15 euros you can bump that up to 300MB speed at home if fibre optic is available.
If all you need is Internet at home then Jazztel are slightly cheaper and let you choose what you want, example you can have Internet and mobile with no landline and you can buy monthly packages example 15 euros for 500MB internet and 500 min calls from your mobile plus 60 minutes to mobiles and free landline calls from your landline.
As far as customer services goes I am sorry to say none of them stand out from the crowd. They are all out to get your money, sell you more than you have, are great at fixing regular issues but if you have something unusual that needs sorting, you'll be stuck with a confused operator on the other end of the line who'll tell you it's not possible even if it is.