At mid 20's and looking for "all things lively" I would imagine that Puerto del Carmen is really the only resort that would definitely tick your box in Lanzarote.
I believe Fuerteventura would be too quiet (haven't actually been there myself, so going on hearsay!) and unless you relish the sort of hedonism that Playa de las Americas (Tenerife) and Playa del Ingles (Gran Canaria) has to offer then perhaps Lanzarote is actually your best option. .... having said that my best memories of fun holidays in my early 20's were in Tenerife & Gran Canaria!!!

Now I've confused myself,and I'm supposed to be giving you sound advice!
January/February/March are probably less reliable months for the guaranteed sunshine that the Canary Islands are famous for, but if you are looking for a "good chance" of sunshine then they are the best option if 4 hours flight time is as much as you are prepared to endure! Dubai/Red Sea or Florida would probably be a safer bet if the sunshine is particularly important to you.
It seldom rains for any length of time in Lanzarote, but from what I have experienced in the last number of years, I would think that the most likely month to get rainfall is actually February. (Sorry!!) That said, we have had lovely holidays in every month over the Spring, so we still visit the island no matter what month it is. We're actually going there from the 29th of February this year for a week, and expect to return to some more freckles than we left with!
Would agree with CG that a heated pool is essential. I am not a "sea creature" so don't really know what the sea is like at any time of the year! And as above, sunbathing is possible when the sun shines, but you may have to seek out places that are well sheltered from the prevailing winds if you want to get down to your itzy bitzy!
When all is said and done, it really depends on just how "LIVELY" you want your holiday to be. Lanzarote is my favourite island but I don't think it would have been my first choice 30 years ago. If I'm to be honest, I'd be suggesting you go for the hedonism, and Playa Del Ingles would be my choice over PDLA.
Oh to be mid 20's again, and mooching about for the craic! You lucky young ducks!

Happy Holidays
From A Knackered Old Bird