Alicante for three/four weeks

It is based on all of my Norwegian extended family’s yearly trips there to get away from winter. Not much help to you on the rental front as some have bought condos there but don’t rent them out.

It would definitely not be my choice, not even for 1 week. It's nowhere near Granada, Seville or Cordoba, the Big 3 destinations in southern Spain (which has the best weather at that time of year). Alicante is 2 to 2-1/2 hours from Valencia by train. I think Valencia itself would be a better choice than Alicante, though possibly a bit cooler, but 3 to 4 weeks in Valencia would drive me mad. Spain is a fabulous country with dozens (probably hundreds) of worthwhile destinations, but it is very large, and the most intriguing destinations are scattered from north to south and from east to west. I wouldn't want to spend all that time in any one place, not even Barcelona, Madrid or Seville. It would be too frustrating not to be able to see the Alhambra, the Alcazar in Seville, the Mezquita in Cordoba, the fabulous modernista architecture in Barcelona, etc.

If you do end up in Alicante, there's at least one good day-trip option (though I'd rather spend a couple of nights there): Cuenca is about 1-1/2 hours away. Its atmospheric old town is on top of a hill. I imagine it could be rather breezy and chilly in the winter, so I'd choose a nice day.

Let's see what others say.

Even with a rental car to take trips from the city, I think that is too long. There are a few things to see in the area and there are some beach towns you could visit, but that time of year many places in the beach towns could still be closed. To me, with some exceptions, most cities can be seen in a few days. I can’t think of any one place in Spain I’d want to stay for 3-4 weeks. I guess it comes down to what you want to do while you’re there. The city is worth visiting.

Alicante would be fun for 1-2 weeks but a bit limited thereafter. There is not that much to see in the city itself, and nearby day trip options are limited as well.
I agree with Valencia and Málaga as alternative suggestions.

The previous replies presume an expectation that you’ll want to use this destination as a base for broader exploration. If that is the case, Alicante maybe isn’t the best.

If your intent is to spend a quiet month in a smaller city near beaches, it might serve quite well.

Most AirBnBs offer a discount on month’s rental. That or VBRO are where I’d start.

In terms of population, Alicante is about 1/3 larger than Granada and about the same size as Cordoba.