I’ve been up Mount Teide about 4 times and as other people have said, it can get a little chilly up the top so take a gilet, full gloves, arm and knee warmers and a hat. I’ve always managed to get away with carrying most of my stuff in a large saddle bag as I hate carrying too much on my back especially on a long climb like that. If you intend going right to the very top to where the cable car is then I would recommend stopping in Vilaflor to fill up on water. There’s a garage at the intersection of TF-51 and TF-21 and I think it’s got a couple of toilets there as well. I think there’s also a cafe at that intersection and some more as you go on a bit further.
As for training for the climbs, just try sweetspot efforts gradually increasing the amount of time at sweetspot. Not sure if it helps, but I used to tilt my bike upwards to replicate going uphill.
While you’re there, if you intend going on any more rides I would definitely recommend traveling up the east coast on the TF-28. Although there are a few rough patches, this has to be my favourite road in Tenerife. The road just keeps on rolling. I tend to go up as far Fasnia (pr a bit further) and then back.
I’d also recommend the TF-563 and TF-21 to Vilaflor, but do that on a different day to your climb up to the top.
Hope you have a great time!