If you're staying in or near Andorra, here's a recommendation for a day's round trip that should have enough curves to satisfy anybody.
You can find a couple of images of maps showing the roads mentioned at the following locations:
http://www.mobilitat.net/carreteres/basica/images/codifgirona.jpg
and
http://www.mobilitat.net/carreteres/basica/descarrega/arealleida.gif
Fill up with cheap petrol in Andorra then head out via the French exit (Pas de la Casa), which is vastly more attractive than the road in and out on the Spanish side. Once into France take the first road on the right which is signposted towards Porté Puymorens. Then follow signs to Puigcerda, where you'll re-enter Spain. From Puigcerda you should follow the N-152 which you should see signposted towards Ripoll and the Collada de Toses (or Tossers if you prefer).
Somewhere near the village of Urtx you'll see one of those wiggly line signs indicating twisty bits for the next 40km which should give you an idea of what to expect. When you reach the Collada de Toses you'll find a large and ugly hotel, and probably a number of bikers parked up and enjoying the view. Take the next right onto the GI-400, and then the second left onto the BV-4031 heading towards Casteller de N'Hug. This route will take you past some of the outlying slopes of the La Molina ski station, and a little further on if you have a day with good visibility you will get tremendous views of the rest of Catalunya looking away towards Barcelona. Further down this road, and past the village of Castellar de N'Hug, you will find the source of the Llobregat river (which flows out to sea just south of Barcelona), where there are a couple of tacky gift shops and a bar where you can sit outside and take a rest. From there you head on to the town of La Pobla de Lillet (bars and restaurants) and take the B-402 heading towards Guardiola de Berguedà.
At Guardiola you have a few options. If you've had enough twisty bits to be going on with or if SWMBO is repeatedly punching you in the kidneys, you can take the C-16 to the Cadí tunnel (toll of about 9 euros) which will bring you back to Puigcerda from where you can return to Andorra via France, or equally it will take you to La Seu d'Urgell from where you can return to Andorra via the Spanish border.
However, if you feel like you can deal with another couple of hours of twisty bits, then a good and lesser-known route is to turn left onto the C-16 heading towards Barcelona for about 1km, then turn right onto the B-400 signposted towards "Massis de Pedraforca", Saldes, and Gósol. This is a wonderful road (especially beautiful in the Autumn) which takes you past Pedraforca, a spectacular twin-peaked limestone mountain which dominates the whole area. Follow this road to the village of Gósol (notable because Picasso stayed there for a while) where you'll find a couple of bars if you want to take a rest. Then head through Gósol on the C-563 to Josa de Cadí and on to Tuixén (try not to have your final drive fail on you on this stretch, because you're really in the middle of nowhere and mobile phone coverage is a bit iffy). At Tuixén, without turning off into the village, head along the same road (it eventually becomes the LV-4008) and after about 30 - 40 km (if I remember rightly) you will end up at La Seu d'Urgell, from where you can head back to Andorra. Keep your eyes open for rocks on the road from Josa to Tuixén.
A third variation you could try from Guardiola is to take the C-16 as far as Bagà, then turn off and take the BV-4024 to the Coll de Pal. This road gives you spectacular curves and views for twenty or so km (watch out for flocks of goats blocking the road) and then the asphalt peters out and it becomes a dirt track which, after a few more km leads down to La Molina ski station, from where you can head back to Puigcerda and Andorra. The best time of year is late spring and summer, otherwise it can get pretty muddy.
As far as Andorra itself is concerned, from about the end of April to the end of October you can often find excellent cheap offers in ski-resort hotels. For example there's an A-Hotel in the village of El Serrat (near the Ordino-Arcalis ski station) where I've spent a couple of weekends with the wife for 45euros for the both of us, half board. This is a good option if you're after peace and quiet. I've got a link to a webpage where you can find these offers which I'll post if I can find it. The Ordino-Arcalis valley is well worth a visit, incidentally, even though the ski station grows larger and uglier each year. If you're around that area, the road from Ordino to Canillo via the Coll d'Ordino is well worth taking. If you're feeling energetic you can park up at the Coll and in a couple of hours walk up the Pic de Casamanya, from the summit of which you get unbeatable panoramic views of the whole of Andorra.
Hope all this gives you some ideas.