Is it safe to travel by bike between cities in Spain?

A Spanish system which approximates your "bicycle roads" is called Vias Verdes. A few of the routes are more than 50km long, but as you can see on their map they're not especially helpful for your route, with the possible exception of the section Sierra de Alcaraz to Albacete. (They're mainly if not entirely replacements for disused railway lines, and the Spanish railway system, like many others, is principally designed around radial connections to the capital).

There are other bicycle lanes, but I'm not aware of any site which aims to map them all.

Is it safe?

No, not at all. It's been already said in other comments and answers, but I feel it has not been stressed enough: as of today, Spain is NOT adapted to cyclists.

• On the road:
  • 99,9% of the road network does not have any separate lanes or roads for bicycles, period.
  • Most roads with big enough shoulders are forbidden to bicycles.
  • Secondary roads usable by cyclist don't have proper shoulders and you have to use the same lane than cars and trucks. You'll even need to leave the road from time to time to give room for tractors etc. which are also forced to use the side of the road.
• On cities:
  • Cities' bike lanes are usually just painted on the sidewalk, no physical separation from pedestrians whatsoever.
  • Most bike lanes on sidewalks were added many many years after the sidewalks were built, with no planning at all, so:
    • you might find trees, light posts, road signals etc. blocking part of the lane;
    • they tend to be short and you'll have to make extensive use of pedestrian crossings and sidewalks to get from one bike lane segment to the next.
  • Pedestrians are legally allowed to, and will use the bike lane for jogging, running, baby carriages... or just because they feel like it, and there's nothing you can do about it. Most will let you pass if you use e.g. your bicycle bell, but still.

Finally: at least 21 cyclist were killed by cars in Spain just in the first 5 months of 2017. That's almost one cyclist killed on the road every week.

So no, not safe. Rather the opposite of safe.

EDIT (as per the comments below): I'm not saying Spain is any more dangerous for cyclists than other countries. I don't have info about other countries. I'm just saying that Spain is, objectively, not safe: it lacks the infrastructure, it lacks the regulation, and it lacks the culture for cyclists to be able to ride safely on roads and cities.

EDIT 2: A couple more facts.
According to a 2009 study on cyclists' safety made by RACE (RACE is Spain's equivalent to Netherlands' ANWB or UK's AA):

• 80% of cyclists said good roads don't exists;
• 72% said cars don't keep legal safe distance of 1,5 m.;
• 56% said cars don't slow down when overtaking them;
• 44% said cars don't respect right of way of cyclists;
• 99% said car drivers' attitude must change for cycling to be safe;
• and finally, 72% of Spanish cyclists declared to find themselves in danger frequently or very frequently while riding.

Safe is never absolute, and with proper planning I'd do it. I've ridden quite a lot in the UK and much less in Spain, so would be cautious to start with (even between different cities a few hours ride apart in the UK the culture of drivers around cyclists varies -- as does cyclists' behaviour/culture).

A couple of tools that may be useful to you:

• Strava's route builder (you'll need to log in but it's free and not onerous) has the setting "global heatmap" to show which routes are popular with cyclists, it also has an option to "use popularity" and automatically route along commonly-ridden roads. I've found it to be unresponsive when trying to route more than about 100 km, so you'll need to apply some manual planning (possibly using Google routes to pick which towns you'll pass through)

• OpenCycleMap shows official routes and cycle paths. This is based on OpenStreetMap (i.e. crowd-sourced) data. Here in the UK that's very good for cycling, but I'm less sure of the situation in Spain.

Statistics are collected on cycling deaths by the European Commission.

https://ec.europa.eu/transport/road_safety/sites/roadsafety/files/pdf/statistics/dacota/bfs20xx_dacota-swov-cyclists.pdf

From this we see many, many things including that Spain has a fairly low rate per million population but its deaths seem to be concentrated less in urban areas than the EU norm. Of course what these figures do not adjust for is the number of journeys made by bicycle in each country.