It's both--there is a historic core surrounded by a large modern city.
I would first map our those day trips with google maps or viamichelin. You may fined them too far your your taste. I quite enjoyed Malaga because I flew in there and out of another city, but I would not necessarily choose it for a base if I was taking a driving trip.

The old city has charm, but it is highly overbuilt, something like Miami Beach. And no, it would not be a good base for doing any day trips beyond of the white villages. I would do the Pueblos Blancos from Ronda, and Úbeda, Baeza and Priego de Cordoba from Jaén or split my time between Úbeda and Cordoba.

Málaga city is genuine Andalucian atmosphere, great tapas bars, cafés, terraces, restaurants, nightlife and 3000 years of history. Within 300 meters in the city centre you'll find the newly excavated Roman theater just beyond the 8-11 century Moorish Alcazaba, the 16th century cathedral, Picassos birth place, the Picasso museum and the new Carmen Thyssen museum.

Be sure to stay close to the city centre. A couple of recent articles about one of my favourite cities in the country: http://www.telegraph.co.uk/journalists/nigel-tisdall/8773310/Malaga-Spain-Secret-Spanish-pleasures.html
http://www.guardian.co.uk/travel/2009/jun/21/malaga-spain-city-breaks-travel?INTCMP=SRCH

Antequera could be another great base, just some 30-40 mins driving North of Málaga city. Known as "the heart of Andalucía" or "the crossroads of Andalucía". Almost right in the middle of the Sevilla, Córdoba, Granada, Málaga rectangle. Here you'll find dolmens (some 5000 years old burial chambers), Roman baths, a Moorish Castle, Gothic churches, Renaissance fountains and Baroque bell towers. Antequera was the first of the Granada emirate towns to fall to the Christians in 1410.
http://www.andaluciacoastandcountry.com/antequera.html

I love the terrace in the corner of the peaceful square in front of the Real Colegiata de Santa María la Mayor, the first partly Renaissance-style church in Spain (built 1514-1550). Up here it's totally peace and quiet. Next to the recently excavated Roman baths beneath and looking up on the 13th century Moorish alcazaba/fortress. Great views over unique landscapes (the spectacular Torcal limestone mountain etc.). http://www.wild-about-travel.com/2011/07/5-reasons-visit-enchanting-antequera/

... agree that Málaga city is too far away from daytrip to Ubeda/Baeza, but Priego de Córdoba and small and not that often visited white villages such as Iznájar and Ardales are within reach. Even closer to Antequera.

Iznájar: http://www.andalucia.com/province/cordoba/iznajar/home.htm
Ardales: http://www.andalucia.com/province/malaga/ardales/home.htm

We loved Malaga for a short visit. But make sure you stay in downtown area. It is a bit far away from "white villages" to use as a base.

Málaga is also very close to Frigiliana (near Nerja), a very pretty village.
http://www.turismofrigiliana.es/es/

And another vote for kimhe's suggestion to visit El Torcal/ Antequera. Very unique landscape.

And, finally, Málaga city is a very pretty town (the historic center), IMO. It obviously lacks the massive attraction of Sevilla's cathedral, but also the hordes of tourists. As kimhe wrote, it feels more genuine - and not like a living museum with more souvenir shops than inhabitants.

As Málaga is not a small town, you should check the exact location of the potential timeshare, though. As Spain is very well covered by Google's Streetview you can get a good idea of how the timeshare's neighborhood looks like if you know the address.