In general the north side of the island is less built up than the south.
It was the first part to get tourism back when the tourists were very posh and chic. Then it got overtaken by flashier developments in the south. So now it has a kind of faded grandure which is very likable.
Its cheaper too.
I went to Puerto de La Cruz and loved it.
It has a charming old town, a nice black sand beach, a cute little harbour.
The Playa Jardin is a really nice ornamental garden- that's worth a visit.
Pp have mentioned the famous artist Cesar Manrique, who built cool things all over the Canary Islands.
His contribution to Tenerife is in Peurto de la Cruz. A concrete sculpture park/seawater pool called Lago Martiánez. Its very cool and you can hang out there all day for a very reasonable entrance price.
I found an adorable brutalist church on the way to the bus station one day but it doesn't seem to be any kind of tourist attraction because the internet has never heard of it! Sorry that's not much help! But just to underline that the architecture is interesting and varied. Not boring high rises!
The great thing about Tenerife is, because it's an island there's just the one road that goes around in a circle. So it's super easy to get to stuff on other parts of the island. Buses are cheap and reliable. Its worth doing the Tede volcano.
And the Waterpark if you have kids with you.
If you want to take a look at the craziness of the main strip (if only to remind yourself why your not staying there) it's waaayy over to the south in Playa de Las Americas.
Again, just hop on a bus.