I'm not fully qualified to make a recommendation as I've never been to Tenerife, although I've been to Lanzarote and Gran Canaria. From what I've head, Tenerife is more like Gran Canaria, which means I have no interest in going there. Ultimately though, it will depend on what you want out of a holiday.
Lanzarote
I fucking love Lanzarote! Everything is absolutely pristine and perfect. I never saw a single pothole (except on the dirt track to the Papagayo beaches of course). All of the tarmac looks like it was laid yesterday; you won't see any litter; everything is in excellent condition. All of the buildings are in a good state of repair and look like they were painted white last Tuesday.
If you stay in a peaceful area (Las Colorades was perfect for me), you can really chill and unwind. I rented a private villa with its own large, heated pool for a similar price to a Jet2/Tui prison camp complex holiday.
There aren't as many of the horrible Brit-filled mega hotels on Lanzarote, perhaps partly because there are significant building restrictions across the island, which is one of the things that makes it pretty much paradise. You can go to a beautiful beach that is not surrounded by a row of giant hotels spoiling the landscape.
Another thing I noticed is that there are no adverts. No billboards or other garish adverts in your face. Nobody trying to sell you anything. Buildings are generally one or two stories high. There are open spaces.
It also has a great vibe. People drive around the quiet streets at about 15mph which was weird for me at first (oh yeah, you can rent a car really really cheaply) but it's just because everyone is so relaxed; nobody is in a hurry; it's just a beautiful, chilled lifestyle. Free parking everywhere. Just park where you want.
Oh and the submarine trip was one of my best-ever life experiences!
All of the above applies if you stay away from Brits, which is pretty easily done. There are only about three of the tacky 'resort' areas full of drunken idiots and they're relatively small here anyway. So I would avoid Playa Blanca and Costa Teguise (and Arrecife - it's like driving in the middle of Manchester with the one-way systems etc.). Then you can have a peaceful time if that's what you want. Lanzarote definitely isn't as much of a party island as the others.
There are many beautiful beaches but they do get busy. Everyone talks about Papagayo as though it's some kind of special secret that nobody knows about (probably cos there's a dirt track road that the hire car companies officially don't let you drive down) but that's bullshit as it gets completely mobbed.
I would fly to Lanzarote today if I could.
Gran Canaria
After being so impressed with Lanzarote, I thought I would try another Canary Island. I was pretty disappointed with Gran Canaria if I'm honest.
There were some really good things about it, for example I stayed in a private villa (again with pool, roof terrace etc.) that was literally on a banana plantation in the north of the island. That was an amazing experience! I spent a lot of time literally just chilling in the middle of the plantation among the banana plants and made good friends with the little dog that lives there.
However, most of the beaches etc. are in the south of the island so it was about an hour's drive to get there. I didn't like the beaches and the resorts. Way too full of drunk Brits, very busy beaches including nudist beaches which freaked my kids out (we ended up there by accident!). Everyone bigs up the Maspalomas sand dunes but I didn't really see anything to get excited about. The beaches aren't even that good.
Gran Canaria does not have the chilled, relaxed vibe of Lanzarote. There are literally 5-lane motorways where everyone drives like a ****. I didn't feel very safe cos of people constantly undertaking and cutting each other up etc. It doesn't make for a relaxing holiday.
Anyway, like I say, I can't comment on Tenerife but I gave the above comparison because Tenerife sounds like it's more akin to Gran Canaria. A well-travelled friend of mine said he would never go back to Tenerife with his family because there was a lot of crime and they didn't feel safe.