Tenerife is an elongated island and it can be walked across by following several routes on the east side. I have walked them myself many times.
The most direct walk from coast to coast across Tenerife island is from Punta del Hidalgo to Santa Cruz, or the other way round.
This route is about 21 kilometers long. The highest point of the route is approximately in the middle of the route, at Cruz de Taborno, 1020 meters above the sea. You will need around 7 hours for the tour.
All other routes across the island are either longer or not for tourists.
Note that some of the routes described below are through highly protected areas, so you need permits for some of them. All the details are available in my text about Anaga mountains.
There are rangers walking around, I have seen them a few times, and if they find you without a permit in a protected zone, you will pay several hundred euros. Note that permits are free, so no need to risk if you do not have to, but there are limits per day.
For the rest of the text, you will need a zoomable map that is provided below. So try to locate the places mentioned in the text:
From Santa Cruz to Punta del Hidalgo, or vice versa
This route starts from the sea level and you climb to 1000 meters above the sea. So this is far from easy, but the route is beautiful and you will see a bit of the rain forest which is mainly in this east side of the island.

It does not matter from which side you start. You can go by bus to Punta del Hidalgo which is on the north side of the island, and start walking towards the south. The picture above shows the cliffs on the north side where you start.
The end of the route will be in the east suburb of the main city Santa Cruz.
I have walked the route both ways, and I have walked it also many times half-way only, this mostly from the south side. You do not need permits for this route.
From Almaciga to Igueste de San Andres, or vice versa
This route is more to the east as compared to the previous route described above. It is a bit shorter but more difficult because of navigation issues.
Note that the highest point on this route is around 880 meters above the sea level. Here just to stress that Almaciga is on the north side (see the zoomable map above) and Igueste is on the south side, but this is all the eastern part of the island.
This route is not advised because it is very difficult to find the way. If you go without a guide it is unlikely you will manage it. There are no marks, you are not supposed to be there in any case. I did it a few times and it was complicated to find the way even for me.
It goes directly through the rain forest, and this is also a highly protected area. You need a permit for this. The picture below is from this route.

It requires around 6 hours minimum, very likely you will need far more. It does not matter from which side you start, in both cases you start from the last stop of a buss line so you can get back to the cities from any of the two ends.
From Almaciga to Chamorga and to Igueste de San Andres, or vice versa
This is a very long tour because it does not go directly across the island. But you will see the most beautiful areas of the east side of the island, from its both coats, and the remote Chamorga village (the picture below). Check the map above to find Chamorga, it is on the east side, the most remote village on the island.

One part is described in my text from Almaciga to Benijo and Chamorga. While writing this text, I realized that I have no description of the second part which I have walked also several times in both directions.
From Chamorga to Pico del Ingles and Santa Cruz, or vice versa
This is a very long tour from the easternmost point of the island. I do not have exact number regarding kilometers but it should be around 36 kilometers or so. There are sub-variants of the route so it is a bit hard to give some precise numbers.
You can go by bus from Santa Cruz to Chamorga and then start walking to the west. This will be through the most attractive areas of the rain forest and there are sections that require a permit. The part from Chamorga to Pico del Ingles is described in my text about a long walk from Chamorga to La Laguna.

The part from the Pico del Ingles to Santa Cruz is described in my mountainnering site, it is just that the walk is described in the opposite direction. The picture below is taken below Pico del Ingles.

Is it easy to get around Tenerife?
The answer is yes. All the tours mentioned above are on the east side, and you can get to the start points by public transportation. No need for taxis or to rent a car. You will want to have a car for other purposes, but not for these tours. These are not round tours so if you go by car, somebody must take it back.
So these are some basic facts you will need if you want to walk across Tenerife. The links given in the text above provide all the details and information. I have walked them all, and many more, so let me know if you need more details.
For these tours you need some hiking shoes or boots, any will do, just note that some parts may be very wet, this is a rain forest area.
From these pictures you realize how this side of the island is different from the Teide area which is described in several of my texts here in the site.
If you plan to visit Tenerife, you can make reservation through Booking.com company which I have used many times in the past on the island and elsewhere.
There is a comment box below so do not hesitate to comment and to ask if you have questions. Thank you for reading and have a nice day.