What sort of temperature could we expect weather-wise
It can vary, but the weather should be mostly good. Maybe identify a few things you can do if it's not 'lazing around in the sun' weather, eg go see the Lanzarote volcano park.
YY to a previous suggestion to be prepared for the evenings to be cooler, especially if you want to still eat and drink in outside restaurants and bars. The sea is bearable, we've been a few times in January/February and swum/ snorkelled for a bit.
Would all bars and restaurants be open, or would it be slim pickings Probably at least 3/4 open, lots of choice, it's not high season, but it's not dead either.
Any advice about specific resorts also welcome
To maximise the chance of the best weather, you want to stay in the south of either island, but avoid Las Americas on Tenerife if you aren't party animals. Costa Adeje is where all the nicer AI hotels are and Los Cristianos is more family orientated. If you don't want to go to a 'resort town' best look for a villa outside this area, but then you have to be prepared to drive everywhere. You can get a good idea about the area by looking at Google Streetview.
Lanzarote is mostly quieter, with only Costa Teguise and the new part of Playa del Carmen (east of the Marina) being 'resort town'. Playa Blanca in the south is quieter. We stayed in a small villa in PB that was five minutes walk from the Marina and beach and had a half decent supermarket and nice bakery a couple of minutes walk away.
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